Evaluate $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^2-4}{x-2}}$ by L’Hôpital’s rule

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$\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{x^2-4}{x-2}}$ $\,=\,$ $\dfrac{0}{0}$

D

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{\dfrac{d}{dx}\big(x^2-4\big)}{\dfrac{d}{dx}(x-2)}}$

D

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{\dfrac{d}{dx}\big(x^2\big)-\dfrac{d}{dx}(4)}{\dfrac{d}{dx}(x)-\dfrac{d}{dx}(2)}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{2x-0}{1-0}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize \dfrac{2x}{1}}$

D

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,2}{\normalsize 2x}$

$=\,\,$ $2 \times 2$

$=\,\,$ $4$

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