Evaluate:
[tex] \\ { \displaystyle{ \rm \lim_{x \to1} \left \{ \frac{ {x}^{4 } - {3x}^{2} + 2 }{ {x}^{3} - {5x}^{2} + 3x + 1} \right \} }} [/tex]
[tex] \: [/tex]
Don't Spam ​

Answer:

To Evaluate:

[tex] \\ { \displaystyle{ \rm \lim_{x \to1} \left \{ \frac{ {x}^{4 } - {3x}^{2} + 2 }{ {x}^{3} - {5x}^{2} + 3x + 1} \right \} }}\\ \\[/tex]

Using L'Hospital's Rule

Let f(x) and g(x) be two common functions which are differentiable on an open Interval , at a point a where,

[tex]\\{ \displaystyle{ \lim_{x \to a}{ \rm{f(x)} = { \displaystyle{ \lim_{x \to a}}} \: { \rm{g(x)}} = 0 \: or \: \pm \infty}}}\\ \\[/tex]

Then,

[tex]\\{ \displaystyle{ \lim_{x \to a} \frac{{ \rm{f (x)}}}{{ \rm{g(x)}}} = { \displaystyle{ \lim_{x \to a }}} \frac{{ \rm{f '(x)}}}{{ \rm{g'(x)}}}}}\\ \\[/tex]

As x to 0 , we have:

[tex]\\{ \displaystyle{ \lim_{x \to 1}}} \left( \large\rm\frac{ {x}^{4} - {3x}^{2} + 2 }{ {x}^{3} - {5x}^{2} + 3x + 1} \right) = \frac{0}{0}\\ \\[/tex]

Therefore,

[tex]\\{ \large{ \displaystyle{ \lim_{x \to1}}} \left( \large \rm\frac{ {x}^{4} - {3x}^{2} + 2}{ {x}^{3} - {5x}^{2} + 3x + 1 } \right) = { \displaystyle{ \lim_{x \to1} { \small{\frac{ {4x}^{3} - 6x}{ {3x}^{2} - 10x + 3 }}} = { \small{\frac{4 - 6}{3 - 10 + 3} }} = - \frac{2}{ - 4} = - \frac{1}{2} }}}\\ \\[/tex]

Hence,

[tex]\\{ \displaystyle{ \lim_{x \to1}{\rm{ \left( \frac{ {x}^{4} - 3 {x}^{2} + 2 }{ {x}^{3} - 5 {x}^{2} + 3x + 1 } \right) = { \boxed{ \red{ - \frac{1}{2} }}}}}}}\\ \\[/tex]

Without using L'Hopital's rule:

Factorize the numerator.

[tex]x^4 - 3x^2 + 2 = (x^2 - 2) (x^2 - 1) = (x^2 - 2) (x + 1) (x - 1)[/tex]

Factorizing cubics is usually a bit trickier. But we know x = 1 makes the denominator vanish, so x - 1 is a factor of the cubic. So for some constants a and b,

[tex]x^3 - 5x^2 + 3x + 1 = (x - 1)^3 + a(x - 1)^2 + b(x - 1)[/tex]

Expanding the right side gives

[tex]x^3 - 5x^2 + 3x + 1 = x^3 + (a - 3)x^2 + (-2a + b + 3)x + a - b - 1[/tex]

Then

[tex]a - 3 = -5 \implies a = -2[/tex]

[tex]-2a + b + 3 = 3 \implies b = -4[/tex]

So we rewrite the limit as

[tex]\displaystyle \lim_{x\to1} \frac{x^4 - 3x^2 + 2}{x^3 - 5x^2 + 3x + 1} = \lim_{x\to1} \frac{(x^2 - 2) (x + 1) (x - 1)}{(x-1)^3 - 2(x-1)^2 - 4(x-1)}[/tex]

x is approaching 1 so x ≠ 1 and we can cancel out factors of x - 1 :

[tex]\displaystyle \lim_{x\to1} \frac{x^4 - 3x^2 + 2}{x^3 - 5x^2 + 3x + 1} = \lim_{x\to1} \frac{(x^2 - 2) (x + 1)}{(x-1)^2 - 2(x-1) - 4}[/tex]

The simplified limand is continuous at x = 1, so we can now evaluate the limit by direct substitution.

[tex]\displaystyle \lim_{x\to1} \frac{x^4 - 3x^2 + 2}{x^3 - 5x^2 + 3x + 1} = \frac{(1^2 - 2) (1 + 1)}{(1-1)^2 - 2(1-1) - 4} = -\frac24 = \boxed{-\frac12}[/tex]

What would Y be equal to in the problem 0.5x + 0.6y = 5.4

The average salary for a first-year teacher is $27,989. If the distribution is approximately normal with a standard deviation of $3250. If the z-score is 0.34. What salary corresponds to the z-score

1. What is the intersection of the sets {0, 2, 6, 8, 12} and {0, 3, 6, 9, 12}?




2. Using the number line below to compare the list of numbers. Place the numbers in


increasing order.


-1.5, .3, -2/5, 1 4/5



3. Using the number line below, list the following numbers in increasing order:


1.1, -1.9, .4, -.6


4. What is the set of even numbers between 3 and 13?





5. What is the intersection of the sets {2, 5,8,12,18} and {1, 2, 5, 7, 18}?




6. What is the intersection between these three sets: X = {-5, -3, -1, 1, 3, 5}, Y= {1, 3, 5, 7, 9, 11}, Z = {-5, -1, 1, 5, 9, 13}




7. What is the union of the sets {0, 2, 4, 8, 10, 12} and {0, 3, 12}?




8. Which value of x makes the open sentence true: 2, 3, 4, or 5? 5 + 2 = 3 + 9




9. Which value of x makes the open sentence true: 1, 2, 3? 3 − 10 ? ( + 1) 2 + 2 3 __________





11. In golf, scores are calculated compared to par for the golf course. On a number line, par is 0.




a. Draw a number line and plot each golfer’s score.


b. List the golfers in order from best (lowest score) to worst (highest score).


c. For the coordinate representing Winston’s score, name the set(s) of which it is an element: natural numbers, whole numbers, integers, and real numbers




Golfer Winston Allen Mariota Hernandez


Score 4-under-par 5-under- par 2-over-par 3-over-par

A box is 36 inches long
18 inches wide
And 6 inches deep
How many cubic feet are in the box

a container holds 24 pints of lemonade. how much is this in gallons?​

Describe how the graph of y = -2(x-4)3 + 1 can be obtained from the graph of y = x3.

What is the quotient of 534÷112

the diameter of a circle is 22.4cm find it's area
​

Of the 14 clients at Bob's salon, 7 of them have red hair. What is the probability that a randomly selected client at Bob's salon has red hair? P(red) =

Find the interest on 2180 for one year at simple interest 4%​

Given the functions, f(x)=30x4+9x3+27x2−21x−17 and g(x)=6x+3, divide f(x) by g(x) and simplify.

Question options:

a.5x3−x2+5x−6+16x+3

b.5x3−x2+5x−6−36x+3

c.5x3−x2+5x−6+66x+3

d.5x3−x2+5x−6+86x+3

which of these expressions or phrases can be used to represent the model brainly choose al the correct answers.

Carmen and her friends go out for lunch and leave a 20% tip. How much was the bill if they left a $17.64 tip?

a plane flying at a certian altitude is observed from two points that are 3 miles apart. the Angles of elevation made by the two points are 55deg and 72deg​

There are 34 desks in the classroom, however, 3 of the desks are broken. How many groups of 4 unbroken
desks can be made?
6 groups
07 groups
8 groups
9 groups

What is the perimeter of STU?
11
104
Time
elapsed
U
00
6
00
MIN
4
SmartScore
out of 100
2
62
-10
-8
-6
-4
0
6
10
s
4
s
т
-2
-4
-6
-8
-10
Perimeter =
units

Select the correct answer.
Selim Is ordering concrete spheres to place as barriers in the city park. The spheres cost $2 per square foot and Selim can spend $20 per
sphere. What is the maximum diameter of the spheres he can purchase?
OA 1.78 ft
ОВ.
1.59 ft
OC. 0.89 ft
OD. 3.56 ft
Reset
Next
2 Edmentum. All rights reserved.
INTL 11:41

HELP PLEASE<33 IM NOT SURE HOW TO DO THIS!

if f(x) = x + 7 and g(x) = 1 / x - 13, whaf is the domain of (f°g)(x)?​

Given the parent function f(x)= ???
in red, write the equation for the transformed function g(x)
in green in terms of f(x)
.

by name and label all inputs. 1: Saving for Retirement Devon is young and has just graduated college with her associate's degree. She plans to start saving money for retirement as soon as she starts her new job. By taking 10% of her monthly gross income, Devon is able to contribute (a) $ $587 each month to a retirement plan. The account is expected to earn interest with an APR of 5.25% compounded monthly. Round answers to two decimal places. a. How much money will be in Devon's retirement account if she continues to make the same monthly investment for 40 years? (3 pt) I b. Overall, Devon contributed how much of her own money into the retirement account? (4 pt)

​

A worker in the USA is paid $22,000 per year as a gross salary. The worker is asked to reduce her salary to $19,000 per annum. What is the percentage decrease? Round you answer to the nearest whole percent.

How do I find the value of x??

If 1 in = 25.4 mm, how fast was the San Rafael Glacier moving in yards per hour? If necessary round to six decimal places

-2w - 5w =

Please help me it’s due tomorrow

What is the measure of BAC. Degree of 53

---
What is the range of the following numbers?
12, 20,18,25,6

A. 5
B. 16 1/5
C. 18
D. 19
E. 25

8. An adult male elephant can weigh up to 15,000 pounds. How many tons is this?
show your work.

help me please!!!!!!!​

( 1 + 1 ) + ( 1 + 1 ) in standard form