help please
simplify the following using L'Hôpital's Rule:
[tex]\displaystyle\lim_{x\to 2}\frac{x^2+x-6}{x^2-4}[/tex]
[tex]\text{note:explanation is a must}[/tex]​

Answer:

[tex] \displaystyle \: \frac{5}{4} [/tex]

Step-by-step explanation:

we are given a expression

we are said to solve it using L'Hôpital's Rule

recall, L'Hôpital's Rule:

[tex] \displaystyle \lim _{x \to \: c}( \frac{f(x)}{g(x)} ) = \lim _{x \to \: c} \frac{f ^{'}(x) }{ {g}^{'}(x) } [/tex]

it is to say the ' means derivative

our given expression:

[tex]\displaystyle\lim_{ x\to 2}\frac{x^2+x-6}{x^2-4}[/tex]

let's apply L'Hôpital's Rule

[tex]\displaystyle\lim_{ x\to 2}\frac{ \dfrac{d}{dx} (x^2+x-6)}{ \dfrac{d}{dx} (x^2-4)}[/tex]

some formulas of derivative

[tex] \displaystyle \: \frac{d}{dx} {x}^{n} = {nx}^{n - 1} [/tex]

[tex] \displaystyle \: \frac{d}{dx} {x}^{} = 1[/tex]

[tex] \displaystyle \: \frac{d}{dx} {c}^{} = 0[/tex]

[tex] \sf \displaystyle \: \frac{d}{dx} {f}^{} (x) + {g}^{}(x) = {f}^{'} (x) + {g}^{'}(x)[/tex]

use sum derivative formula to simplify:

[tex]\displaystyle\lim_{ x\to 2}\frac{ \dfrac{d}{dx} (x^2)+ \dfrac{d}{dx} (x) + \dfrac{d}{dx}( -6)}{ \dfrac{d}{dx} (x^2) + \dfrac{d}{dx} (-4)}[/tex]

simplify using exponents using exponent derivative formula:

[tex]\displaystyle\lim_{ x\to 2}\frac{ 2x+ \dfrac{d}{dx} (x) + \dfrac{d}{dx}( -6)}{ 2x + \dfrac{d}{dx} (-4)}[/tex]

use variable derivative formula to simplify variable:

[tex]\displaystyle\lim_{ x\to 2}\frac{ 2x+ 1+ \dfrac{d}{dx}( -6)}{ 2x + \dfrac{d}{dx} (-4)}[/tex]

use constant derivative formula to simplify derivative:

[tex]\displaystyle\lim_{ x\to 2}\frac{ 2x+ 1+ 0}{ 2x + 0}[/tex]

simplify addition:

[tex]\displaystyle\lim_{ x\to 2}\frac{ 2x+ 1}{ 2x }[/tex]

since we are approaching x to 2

we can substitute 2 for x

[tex]\displaystyle\lim_{ x\to 2}\frac{ 2.2+ 1}{ 2.2 }[/tex]

simplify multiplication:

[tex]\displaystyle\frac{ 4+ 1}{ 4}[/tex]

simplify addition:

[tex] \displaystyle \: \frac{5}{4} [/tex]

hence,

[tex]\displaystyle\lim_{ x\to 2}\frac{ \dfrac{d}{dx} (x^2+x-6)}{ \dfrac{d}{dx} (x^2-4)} = \frac{5}{4} [/tex]

Answer:

5/4 is your answer hope it helps

PLEASE I NEED URGENT HELP!!!!! lan and Jessica each save their quarters. Ian starts out with 34 quarters and saves 8 quarters a month. Jessica starts out with 2 quarters and saves 16 quarters a month. In how many months will Jessica have the same number of quarters as lan? How many quarters will each of them have?

Noah randomly selects one of his eight different pairs of shoes to wear each day. Of his eight pairs of​ shoes, Noah has two pairs of boots and one pair of loafers. For how many days of the next 264 is it expected that Noah will wear either boots or​ loafers?​

A contractor rents a trailer truck to
haul materials. The inside of the
trailer is 20 feet long, 7 feet wide,
and 8 feet high. Find the storage
capacity of the trailer in cubic feet.

What is the closure property of multiplication for rational
expressions? Explain why rational expressions are
closed under multiplication.

Alexa is driving on the freeway. She passed mile marker 400. The fastest she will drive is 65 mph, though there is a bit of traffic that might slow her down a bit, but she doesn't know exactly how much. The mile markers are increasing on her route. Write an equality that represents alexa’s speed

How to factorise

a²b² - 6abc + 9c²

can anyone explain it by step-by-step
pls ​

1. Which equation represents a line that is parallel to the line y = 4x – 1?

6. Place the decimals in order from
least to greatest.
2.12 2.21 2.14 2.41

4.51 45.1 54.1 4.15 41.5

The average temperatures for two of the world's hottest places are given. Determine which location has the warmer average temperature and by how many degrees fahrenheit. Location A: 86.7 F and Location B: 30.1 C

if a mom has 6 kids and i kidnapped 2 how many kids does she have left? ;) just asking

Leslie has a box of markers. There are 3 black,
4 red, 2 blue, 1 brown, 2 purple, 2 yellow and 5
green markers. Create a sample space of possible
outcomes for choosing a marker. Is this a simple or compound event?

Find the area of the following trapezoid.
14
7
5

A triangle has vertices on a coordinate grid at L(9,7)L(9,7), M(9,-8)M(9,−8), and N(1,7).N(1,7). What is the length, in units, of \overline{LM} LM ?

Marcus has 3/4 of and 8-piece pizza left over from last night. He eats 1/3 of the left over pizza for a snack. How much of the original pizza is left after his snack?​

what is 2y-x=-4 in slope intercept form

The triangle shown has an area of 32.5 square centimeters. Find the measure of the base (segment XY)

y = 100 - 8.9x percent

An item is regularly priced at $30. It is now priced at a discount of 30% off the regular price. What is the price now?

helppp ! please and thank you

Cups are sold in packages of 8. Napkins are sold in packages of 12.

What is the fewest number of packages of cups and the fewest number of packages of napkins that can be purchased so there will be the same number of cups as napkins?

Natalie had 20 lbs. of cat food. If she feeds
her cat 6 of a pound daily, how many days
will the food she has last?

у = - 2х
y = 8x — 10

true or false: the diagonals of a rectangle bisect opposite angles​

Please help with #28 I’m really struggling to figure this answer out!

Evaluate ba+cd if a=7/8 , b=−7/16 , c=0.8 , and d=1/4 . Write your answer as a mixed number in simplest form.

HURRY NEED HELP ASAP


The following information is known about a loan.

Time = 8 years
Interest rate = 14%
Principal = $1,010

What is the total amount of simple interest that is earned?
$113.20
$1,131.20
$11,311.20
$113,112.00

-30=p-17
help pleaseee

What is the approximate diameter of a tire that will make 775 complete rotations in one mile? (Note: there are 5280 feet
in 1 mile)

2/5p-3.11=1.13
what is p

this is really easy i just dont get it