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

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