Answers
Explanation
Given
[tex]w^3-1000=0[/tex]We will have;
[tex]\begin{gathered} w^3=1000 \\ \mathrm{For\:}x^3=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt[3]{f\left(a\right)},\:\sqrt[3]{f\left(a\right)}\frac{-1-\sqrt{3}i}{2},\:\sqrt[3]{f\left(a\right)}\frac{-1+\sqrt{3}i}{2} \\ therefore;\text{ }w=\sqrt[3]{1000},\:w=\sqrt[3]{1000}\frac{-1+\sqrt{3}i}{2},\:w=\sqrt[3]{1000}\frac{-1-\sqrt{3}i}{2} \\ hence;w=10,w=10\times\frac{-1+\sqrt{3}i}{2},\:w=10\times\frac{-1-\sqrt{3}i}{2} \\ w=10,\:w=-5+5\sqrt{3}i,\:w=-5-5\sqrt{3}i \end{gathered}[/tex]Answer: Option D
Related Questions
Graph y-12=4(x-(-3))
Answers
The graph of the equation can be given as
What is an equation of the line how to graph the equation?
An equation is an expression between two variables With an equality sign. Usually an equation of the line can be given as y= mx + c, Where x is independent and y is dependent and c is y-intercept. as the value of x changes the value of y also changes.
We are given an equation y-12 = 4(x-(-3))
We first simplify the equation
we get y -12 = 4(x+3)
Multiplying 4 inside the bracket we get,
y-12= 4x+12
Adding 12 on both sides to get slope intercept form.
y=4x+24
The slope of the line is 4 and y-intercept is 24.
Now we can plot the given equation.
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What is the location of the vertex on the parabola defined by f(x) = 2x2 + 14x + 11 and in what direction does the parabola open?
Answers
Vertex of a parabola
The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved.
The given function is,
[tex]f(x)=2x^2+14x+11[/tex]We will apply graphical method to obtain the solution to the vertex of the parabola
From the graph above, the vertex of the parabola is (- 3.5, - 13.5) which can also be represented into fraction as
[tex](-\frac{7}{2},-\frac{27}{2})[/tex]And finally, the parabola opens upward.
Hence, the correct option is Option 2.
Can you please help me
Answers
Step 1
A rectangle is a quadrilateral with the parallel opposite two sides being equal and with each vertex being 90°. The sum of angles in a rectangle is 360°. It has 2 diagonals.
Step 2
Derive an answer from the statement in step 1
The consecutive sides are perpendicular is the only right option.
Hence option D
can anyone give me an example of plotting points with a real life word problem
Answers
Let's have an example of a seller that sells phones.
His salary have a fixed part of $50, and it increases by $2 by each phone he sells.
We want to find out how much will be his salary if he sells 5, 10 and 15 phones.
If we use the variable x to represent the number of phones sold, his salary (variable y) will be defined by the equation:
[tex]y=50+2x[/tex]Now, using the values of x we want to calculate, we have that:
[tex]\begin{gathered} x=5\colon \\ y=50+2\cdot5=50+10=60 \\ \\ x=10\colon \\ y=50+2\cdot10=50+20=70 \\ \\ x=15\colon \\ y=50+2\cdot15=50+30=80 \end{gathered}[/tex]We can plot these points to see how much his salary increases for each phone sold, and also we can have an idea of any point we want to find out:
Find the first five terms of the sequence
defined recursively as follows: t₁ = 1,
tn = 3(t(n-1)), n≠1, n is a natural number.
Answers
Answer:
1, 3, 9, 27, 81
Step-by-step explanation:
using the recursive rule
[tex]t_{n}[/tex] = 3 [tex]t_{n-1}[/tex] with t₁ = 1 , then
t₂ = 3 × t₁ = 3 × 1 = 3
t₃ = 3 × t₂ = 3 × 3 = 9
t₄ = 3 × t₃ = 3 × 9 = 27
t₅ = 3 × t₄ = 3 × 27 = 81
the first 5 terms are 1, 3, 9, 27, 81
What is the equation of a line that passes through the point (8, -4) and is parrallel to the line 6x + 2y = 9
Answers
The equation of a line that passes through the point (8, -4) and is parallel to the line 6x + 2y = 9 is 3x+y-20=0.
6x + 2y - 9 = 0
2y = -6x + 9
divide by 2 on both sides
y = -3x + 9/2
slope m = -3 and c = 9/2.
Line equation through point(8, -4) is y-y1 = m(x-x1)
y - (-4) = -3(x-8)
y+4=-3x+24
3x+y-20=0.
Therefore the equation of a line that passes through the point (8, -4) and is parallel to the line 6x + 2y = 9 is 3x+y-20=0.
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What is the length of BC to the nearest 10th of a centimeter
Answers
Step 1: sketch out the right angled triangle
Step 2: calculate the value BC
To calculate the value of BC we will use the trigonometric ratio
[tex]\begin{gathered} \sin \theta=\frac{opp}{hyp} \\ \text{where,} \\ \text{opp}=BC \\ \text{Hyp}=6.2\operatorname{cm} \\ \theta=32^0 \end{gathered}[/tex]By substituting the values, we will have
[tex]\begin{gathered} \sin 32^0=\frac{BC}{6.2} \\ BC=6.2\times\sin 32^0 \\ BC=6.2\times0.5299 \\ BC=3.2854\operatorname{cm} \\ \text{appro}\xi\text{mately to the nearest tenth we will have} \\ BC=3.3\operatorname{cm} \end{gathered}[/tex]Hence,
The value of BC =3.3cm
Write a transformation of a quadratic function with a vertical stretch by a factor of 2, followed by a horizontal shift of 3 units to the left and 5 units down.show workkkk!!!
Answers
The standard form of a quadratic function presents the function in the form
[tex]f(x)=a(x-h)^2+k[/tex]where (h, k) is the vertex.
The standard form is useful for determining how the graph is transformed from the graph of y = x^2. The figure below is the graph of this basic function.
You can represent a horizontal (left, right) shift of the graph of
by adding or subtracting a constant, h, to the variable x, before squaring. Here h = -3
[tex]y=(x+3)^2[/tex]The magnitude of a indicates the stretch of the graph. a = 2
[tex]y=2(x+3)^2[/tex]Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply.)
Answers
Solution
The given function is
[tex]f(x)=4x^3[/tex]With given interval
[tex]\lbrack1,2\rbrack[/tex]The function is differentiable on the open interval (1,2) and it is continuous on the closed interval [1,2]
Therefore mean value theorem can be used
Calculating the c value iit follows:
[tex]f^{\prime}(c)=\frac{f(2)-f(1)}{2-1}[/tex]This gives
[tex]\begin{gathered} f^{\prime}(c)=\frac{4(2)^3-4(1)^3}{1} \\ f^{\prime}(c)=\frac{32-4}{1} \\ f^{\prime}(c)=28 \end{gathered}[/tex]Differentiating the given function gives:
[tex]f^{\prime}(x)=12x^2[/tex]Equate f'(c) and f'(x)
This gives
[tex]x^2=\frac{28}{12}[/tex]Solve the equation for x
[tex]\begin{gathered} x^2=\frac{28}{12} \\ x^2=\frac{7}{3} \\ x=\pm\sqrt{\frac{7}{3}} \\ x=\sqrt{\frac{7}{3}},x=-\sqrt{\frac{7}{3}} \end{gathered}[/tex]Therefore the values of c are
[tex]\sqrt{\frac{7}{3}},-\sqrt{\frac{7}{3}}[/tex]If you were to solve the following system by substitution, what would be the best variable to solve for and from what equation? 3x + 6v=9 2x – 10v=13
Answers
The best variable to solve is x=3-2v, after dividing the first equation by 3.
x=4, and v=-1/2
1) Solving that system by Substitution
2) Making then
x=3-2v
2x-10v=13
3) Plugging into the 2nd equation
2(3-2v)-10v=13
6-4v -10v=13
6-14v=13
-14v=13-6
-14v=7
v=-1/2
Plugging into the first equation
x=3-2v
x=3-2(-1/2)
x=3+1
x=4
Write an equation for the line parallel to g(x)= -2x-6 and passing through the point (7, 4) Write the answer in slop intercept form.
Answers
The slope-intercept form is:
[tex]y=mx+c[/tex]So first we will find the gradient:
Parallel lines have the same gradient:
[tex]\begin{gathered} g(x)=-2x-6 \\ \text{The gradient from the equation above is -2} \end{gathered}[/tex]So now that we know the gradient of the line as -2, we will then find the equation of the line using the formula below as it passes through (7,4):
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ \text{From (7,4)} \\ x_1=7,y_1=4 \\ y-4=-2(x-7) \\ y-4=-2x+14 \\ y=-2x+14+4 \\ y=-2x+18 \end{gathered}[/tex]balanced equation question for lrwctoce
Answers
Letter B is the correct answer.
2 Al 2
3 Ni 3
6 Br 6
The other equatins are unbalanced
1)Lin solved the equation incorrectly. Find all the errors in her solution. (Select all appropriate answers.)Given to Step 1: Lin calculated 4 - 17 incorrectly; should be -13.Step 1 to Step 2: Lin distributed 8(x - 3) incorrectly; should be 8x - 3.Step 2 to Step 3: Lin calculated 8x - 24 + 7 incorrectly; should be 8x - 31.Step 3 to Step 4: Lin should have subtracted 8x from each side; should be 18x.What should Lin's answer have been ?8 ( x - 3 ) + 7 = 2x ( 4 - 17 )answer choices :x = 3/2 x = 1/2x = 1/5x = 1/3
Answers
Answer:
Given to Step 1: Lin calculated 4 - 17 incorrectly; should be -13.
Step 3 to Step 4: Lin should have subtracted 8x from each side; should be 18x.
Lin's answer should be x = 1/2
Explanation:
The initial expression is:
8(x - 3) + 7 = 2x(4 - 17)
So, the first error is in step 1, because 4 - 17 = - 13, then step 1 should be:
8(x - 3) + 7 = 2x(-13)
Then, we can apply the distributive property as:
8x - 8(3) + 7 = -26x
8x - 24 + 7 = -26x
We need to add similar terms:
8x - 17 = - 26x
Then, we can subtract 8x from both sides as follows:
8x - 17 - 8x = - 26x - 8x
-17 = -34x
Finally, we can divide by -34 as follows:
-17/(-34) = -34x/(-34)
1/2 = x
Therefore, the answers are:
Given to Step 1: Lin calculated 4 - 17 incorrectly; should be -13.
Step 3 to Step 4: Lin should have subtracted 8x from each side; should be 18x.
Lin's answer should be x = 1/2
Which of the following is an infinite series?A) 3, –6, 12, –24, 48B) 2 − 6 + 18 − 54 + . . .C) 3, 13, 23, 33, . . .D) 4 + 8 + 16 + 32
Answers
SOLUTION
An infinite series is the sum of an indefinitely many numbers related in a given way. That is the addition of such number is continuos. So our answer should be between
B) 2 − 6 + 18 − 54 + . . . and
C) 3, 13, 23, 33, . . .
But, looking at both, C) is a sequence showing ordered list of numbers.
Hence B) is a showing series showing sum of a list of numbers or showing multiplication pattern.
Therefore, the correct answer is option B.
HiThe scatter plot shows a hiker's elevation above sea level during a hike from the base to the
top of a mountain. The equation of a trend line for the hiker's elevation is y = 7.96x +676, where x
represents the number of minutes and y represents the hiker's elevation in feet. Use the equation of
the trend line to estimate the hiker's elevation after 170 minutes.
Answers
Answer:
2029.2 ft
Step-by-step explanation:
If x represents the number of minutes, then all you have to do is plug in 170 ( the number of minutes ) into the equation.
y = 7.96 (170) + 676
y = 1353.2 + 676
y = 2029.2
y represents the hiker's elevation in feet, so the answer would be 2029.2 ft.
let me know if anything is confusing :-))
There can be no more than 100 people in the movie theater. There are already 22 people in the movie theater.What inequality represents the number of additional people, p, that can enter the movie theater?Drag and drop the appropriate symbols to correctly complete the inequality.
Answers
Answer:
[tex]p\text{ + 22}\leq\text{ 100}[/tex]Explanation:
Here, we want to drop the appropriate symbols
When we add the given number 22 to p, it would give a number which is at most 100
That means the number must be less than or equal to 100
mathematically, we have that as:
[tex]p\text{ + 22}\leq\text{ 100}[/tex]48. Mrs. Dalton is selling pieces of cake at the school carnival. She baked 8 cakes and cut them
each into 12 pieces. If she sold 81 slices of cake, how many total cakes does she have left?
Circle the correct option.
Answers
Total number of cakes left is 1.25 i.e. 1 and quarter.
Given,
Number of cakes = 8
Number of pieces of each cake =12
Number of slices sold = 81
Then,
Total number of slices of cake = [tex]12*8=96[/tex]
Now,
number of remaining slices =[tex]96-81=15[/tex]
To find number of remaining cake left, divide number of remaining slices to the pieces of each cake.
Number of remaining cake = [tex]\frac{15}{12} =1.25[/tex]
Thus, total number of cakes left is 1.25 i.e. 1 and quarter.
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Lauren uses1/3cup of carrot juice for every2/3cup of apple juice to make a fruit drink.Enter the number of cups of carrot juice Lauren uses for 1 cup of apple juice,
Answers
1/3 cup of carrot juice ---------------------------------->2/3 cup of apple juice
x cups of carrot juice ----------------------------------->1 cup of apple juice
Using cross multiplication:
[tex]\frac{\frac{1}{3}}{x}=\frac{\frac{2}{3}}{1}[/tex]solve for x:
[tex]\begin{gathered} x=\frac{\frac{1}{3}}{\frac{2}{3}} \\ x=\frac{3}{6} \\ x=\frac{1}{2} \end{gathered}[/tex]She uses 1/2 cups of carrot juice for 1 cup of apple juice
Which of the statements below is true for the following set of numbers? 20, 15, 50, 85, 75, 60A. The range is 70 and the midrange is 35 B.the range is 70 and the midrange is 50C. The range is 85 and the midrange is 55D.the range and the midrange are equal
Answers
B
1) Firstly, we need to orderly write this sequence of numbers:
[tex]15,20,50,60,75,85[/tex]2) Then we need to calculate the Range of this Data Set, by subtracting from the highest values the least one:
[tex]R=85-15=70[/tex]2.2) The Midrange is the average between the Highest Value and the Least value on this Data Set:
[tex]M=\frac{85+15}{2}=50[/tex]3) Thus the answer is B
The Super Discount store is having a sale and all the clearance items are %40 off. Which of the following are correct ways to find the price of an item that is $25.00 normally ?
Answers
The normal price is $25, we need to find the 40% of $25.
40% = 0.4
So if we multiply $25 by 0.4
The table shows the predicted growth of bacteria after various numbers of hours. Write an explicit formula for the number of bacteria after n hours.Hours1 2 3 4 5(n)Numberof 33 57 81 105 129BacteriaΟ Α.a= 24+9O B. a = 9n+ 24OC. a, = 24n+33OD. a = 9n +33
Answers
Let a be the number of bacteria after one hour.
From the table, we get a=33.
The difference between 57 and 33 is 57-33= 24.
The difference between 81 and 57 is 81-57=24
Hence we get the common difference d=24.
The given data is in the arithmetic progression.
The formula for the nth term in the arithmetic progression is
[tex]a_n=a+(n-1)d[/tex]Substitute a=33 and d=24, we get
[tex]a_n=33+(n-1)24[/tex][tex]a_n=33+24n-24[/tex][tex]a_n=24n+9[/tex]Hence the required recursive equation is
[tex]a_n=24n+9[/tex]Option A is correct.
For a given geometric sequence, the common ratio, r, is equal to 5, and the 7th term, an, is equal to -43. Find the value of the 9thterm, a9. If applicable, write your answer as a fraction.a9=
Answers
Given:
it is given that common ration of a geometric sequence is r = 5 and 7th term is - 43.
Find:
we have to find the value of 9th term.
Explanation:
we know the formula for nth term of a geometric sequence is
[tex]a_n=ar^{n-1}[/tex]since, 7the term is - 43,
Therefore, we have
[tex]\begin{gathered} a_7=-43 \\ ar^{7-1}=-43 \\ ar^6=-43 \\ a(5)^6=-43 \\ a(15625)=-43 \\ a=-\frac{43}{15625} \end{gathered}[/tex]The 9the term of the geometric sequence is
[tex]\begin{gathered} a_9=-\frac{43}{15625}\times(5)^{9-1} \\ =-\frac{43}{15625}\times(5)^8 \\ =-\frac{43}{(5)^6}\times(5)^8 \\ =-43\times25 \\ a_9=-1075 \end{gathered}[/tex]Therefore, 9th term of given geometric sequence is -1075.
Recall that the formula for an area of a circle is A=πr^2 Find the radius of a circle whose area is 300 square meters. Give a decimal approximation rounded to 2 decimal places.raduis=_______meters
Answers
Given:
a.) Area of circle = 300 m²
Let's determine its radius,
[tex]\text{ Area = }\pi\text{r}^2[/tex][tex]\text{ }\pi r^2\text{ = Area}[/tex][tex]\text{ r}^2\text{ = }\frac{300}{3.1416}\text{ ; }\pi\text{ }\approx\text{ 3.1416}[/tex][tex]\text{ r = }\sqrt{\frac{300}{3.1416}}[/tex][tex]\text{ r = 9.77203881243 }\approx\text{ 9.77 m}[/tex]Therefore, the radius of the circle is approximately 9.77 m
List each real zero of f according to the behavior of the graph at the x-axis near that zero. Zero(s) where the graph crosses the x-axis:Zero(s) where the graph touches, but does not cross the x-axis:
Answers
In the graph, there are 2 zeros, one for each type.
The zero where the graph crosses the x axis is 1. When x=1, the function intercepts the x axis and crosses it.
The zero where the graph touches but does not cross the x axis is -1. When x=-1, the function touches x axis but goes back to the quadrant.
14. Log(x)=2meansa. x=10b. x=2^10c. x=10^2d. x=0e. none of the above
Answers
the graph to the right represents the cost of a taxi where X is distance in miles and y is cost $10 what conclusion can you make? A. the taxi will stop every two miles B. The taxi cost $2 per mile C. The taxi will stop after 5 miles D. The taxi cost $2 just to get into attacks
Answers
Explanation
Step 1
the graph to the right represents the cost of a taxi where X is the distance in miles and y is the cost.
if the cost is $10m it means,exist a x, that satisfies
[tex]f(x)=10[/tex]look for the y value = 10 in the graph
Step 2
down vertically for to find the x value
BC and DE are chords of circle A, and BC DE. Which statement cannot be verifiedfrom the information that is given?BC DEBZBACZ ZDAEAABC = AADEZDAE CAD
Answers
Let's start in the first information:
[tex]arcBC\cong arcDE[/tex]This can be verified, because, AC, AB, AD and AE are all congruent, because they are the radius of A and, since BC and DE are congruent, triangles ABC and ADE are also congruent.
Thus, the angles mBAC and mDAE are congruent. The measure of thesee angles are the same as the arcs BC and DE, so we verified this alternative.
In the explanation above, we also verified the second alternative:
[tex]\angle BAC\cong\angle DAE[/tex]And we verified the third alternative:
[tex]\Delta ABC\cong\Delta ADE[/tex]We are left with the last alternative:
[tex]\angle DAE\cong\angle CAD[/tex]This can't be verified. One way of seeing that is that we can rotate triangle ABC around the circle without changing any of the given information, however this changes mCAD.
So the only statement that cannot be verified from the given information is the last one:
[tex]\angle DAE\cong\angle CAD[/tex]-4 5/9 + (-1 2/3)[tex] - 4 \frac{5}{9} + ( - 1 \frac{2}{3} )[/tex]
Answers
Answer:
-6 2/9
Explanation:
Given the expression:
[tex]-4\frac{5}{9}+(-1\frac{2}{3})[/tex]First, open the brackets:
[tex]=-4\frac{5}{9}-1\frac{2}{3}[/tex]Next, change the fractions to improper fractions:
[tex]=-\frac{41}{9}-\frac{5}{3}[/tex]Then, take the lowest common multiple of 9 and 3 to combine the fractions:
[tex]\begin{gathered} =\frac{-41-5(3)}{9} \\ =\frac{-41-15}{9} \\ =\frac{-56}{9} \\ =-6\frac{2}{9} \end{gathered}[/tex]The result is -6 2/9.
Find the exact coordinates of the HOLE in this rational function: R (x)=×+1/×+(-1)
Answers
Can you please help me with 28 Please give all end behavior such as limits and as_,_
Answers
We must describe the local and end behaviour of the function:
[tex]f(x)=\frac{x^2-4x+3}{x^2-4x-5}.[/tex]First, we rewrite the polynomials in numerator and denominator in terms of their roots:
[tex]f(x)=\frac{(x-1)\cdot(x-3)}{(x+1)\cdot\mleft(x-5\mright)}\text{.}[/tex]Local behaviour
We see that f(x) has a zero in the denominator for x = -1 and x = 5. The function f(x) has vertical asymptotes at these values. To analyze the local behaviour, we must compute the lateral limits for x → -1 and x → 5.
Limit x → - 1 from the left
Computing the limit from the left when x → -1, is equivalent to replacing x by -1 - ε and computing the limit when ε → 0:
[tex]\begin{gathered} \lim _{x\rightarrow-1^-}f(x)=\lim _{\epsilon\rightarrow0}f(-1-\epsilon) \\ =\lim _{\epsilon\rightarrow0}\frac{(-1-\epsilon-1)\cdot(1-\epsilon-3)}{(-1-\epsilon+1)\cdot(-1-\epsilon-5)} \\ =\lim _{\epsilon\rightarrow0}\frac{(-2)\cdot(-2)}{(-\epsilon)\cdot(-6)}\rightarrow+\infty. \end{gathered}[/tex]In the last step, we can't throw the ε in the parenthesis different to zero.
Limit x → - 1 from the right
Computing the limit from the left when x → -1, is equivalent to replacing x by -1 + ε and computing the limit when ε → 0:
[tex]\begin{gathered} \lim _{x\rightarrow-1^+}f(x)=\lim _{\epsilon\rightarrow0}f(-1+\epsilon) \\ =\lim _{\epsilon\rightarrow0}\frac{(-1+\epsilon-1)\cdot(1+\epsilon-3)}{(-1+\epsilon+1)\cdot(-1+\epsilon-5)} \\ =\lim _{\epsilon\rightarrow0}\frac{(-2)\cdot(-2)}{(+\epsilon)\cdot(-6)}\rightarrow-\infty. \end{gathered}[/tex]In the last step, we can't throw the ε in the parenthesis different to zero.
Limit x → 5 from the left
Computing the limit from the left when x → 5, is equivalent to replacing x by 5 - ε and computing the limit when ε → 0:
[tex]\begin{gathered} \lim _{x\rightarrow-1^-}f(x)=\lim _{\epsilon\rightarrow0}f(-1-\epsilon) \\ =\lim _{\epsilon\rightarrow0}\frac{(5-\epsilon-1)\cdot(5-\epsilon-3)}{(5-\epsilon+1)\cdot(5-\epsilon-5)} \\ =\lim _{\epsilon\rightarrow0}\frac{(+4)\cdot(+2)}{(+4)\cdot(-\epsilon)}\rightarrow-\infty. \end{gathered}[/tex]In the last step, we can't throw the ε in the parenthesis different to zero.
Limit x → 5 from the right
Computing the limit from the left when x → 5, is equivalent to replacing x by 5 + ε and computing the limit when ε → 0:
[tex]\begin{gathered} \lim _{x\rightarrow-1^-}f(x)=\lim _{\epsilon\rightarrow0}f(-1-\epsilon) \\ =\lim _{\epsilon\rightarrow0}\frac{(5+\epsilon-1)\cdot(5+\epsilon-3)}{(5+\epsilon+1)\cdot(5+\epsilon-5)} \\ =\lim _{\epsilon\rightarrow0}\frac{(+4)\cdot(+2)}{(+4)\cdot(+\epsilon)}\rightarrow+\infty. \end{gathered}[/tex]In the last step, we can't throw the ε in the parenthesis different to zero.
End behaviour
To describe the end behaviour of the function, we must compute the limits of the function when x → -∞ and x → +∞.
Limit x → -∞
[tex]\begin{gathered} \lim _{x\rightarrow-\infty^{}}f(x)=\lim _{x\rightarrow-\infty^{}}\frac{x^2-4x+3}{x^2-4x-5} \\ =\lim _{x\rightarrow-\infty}\frac{x^2-4x+3}{x^2-4x-5}=\frac{\lim _{x\rightarrow-\infty}\frac{x^2-4x+3}{x^2}}{\lim _{x\rightarrow-\infty}\frac{x^2-4x-5}{x^2}}=\frac{1}{1}=1. \end{gathered}[/tex]To compute the limit we have divided numerator and denominator by x² and distributed the limit. The result of each limit is given by the leading term, which has the highest power of x.
Limit x → +∞
[tex]\begin{gathered} \lim _{x\rightarrow+\infty^{}}f(x)=\lim _{x\rightarrow+\infty^{}}\frac{x^2-4x+3}{x^2-4x-5} \\ =\lim _{x\rightarrow+\infty}\frac{x^2-4x+3}{x^2-4x-5}=\frac{\lim_{x\rightarrow+\infty}\frac{x^2-4x+3}{x^2}}{\lim_{x\rightarrow+\infty}\frac{x^2-4x-5}{x^2}}=\frac{1}{1}=1. \end{gathered}[/tex]To compute the limit we have divided numerator and denominator by x² and distributed the limit. The result of each limit is given by the leading term, which has the highest power of x.
AnswersLocal behaviour
The function f(x) has vertical asymptotes at x = -1 and x = 5.
[tex]\begin{gathered} \lim _{x\rightarrow-1^-}f(x)=+\infty \\ \lim _{x\rightarrow-1^+}f(x)=-\infty \\ \lim _{x\rightarrow5^-}f(x)=-\infty \\ \lim _{x\rightarrow5^+}f(x)=+\infty \end{gathered}[/tex]End behaviour
[tex]\begin{gathered} \lim _{x\rightarrow-\infty^{}}f(x)=1 \\ \lim _{x\rightarrow+\infty^{}}f(x)=1 \end{gathered}[/tex]