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otez555 [7]
2 years ago
11

PLZ HELP

Mathematics
1 answer:
Lelechka [254]2 years ago
7 0

Answer:

5 quartz would make the most sense, hope this helps

Step-by-step explanation:

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mafiozo [28]

Answer:

\displaystyle \:  \frac{5}{4}

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:

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

it is to say the ' means derivative

our given expression:

\displaystyle\lim_{  x\to 2}\frac{x^2+x-6}{x^2-4}

let's apply L'Hôpital's Rule

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

some formulas of derivative

\displaystyle \:  \frac{d}{dx}  {x}^{n}  =  {nx}^{n - 1}

\displaystyle \:  \frac{d}{dx}  {x}^{}  =  1

\displaystyle \:  \frac{d}{dx}  {c}^{}  =  0

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

use sum derivative formula to simplify:

\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)}

simplify using exponents using exponent derivative formula:

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

use variable derivative formula to simplify variable:

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

use constant derivative formula to simplify derivative:

\displaystyle\lim_{    x\to 2}\frac{ 2x+ 1+  0}{ 2x +  0}

simplify addition:

\displaystyle\lim_{    x\to 2}\frac{ 2x+ 1}{ 2x }

since we are approaching x to 2

we can substitute 2 for x

\displaystyle\lim_{    x\to 2}\frac{ 2.2+ 1}{ 2.2 }

simplify multiplication:

\displaystyle\frac{ 4+ 1}{ 4}

simplify addition:

\displaystyle \:  \frac{5}{4}

hence,

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

7 0
3 years ago
Read 2 more answers
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