Question

If G(X)=-25-x2 then limx - 1G(x)
-G(1)x-1 has the value of

​

Answer 1

$\textbf{Given:}$

$\mathsf{G(x)=25-x^2}$

$\textbf{To find:}$

$\textsf{The value of}\;\mathsf{\displaystyle\lim_{x\to\;1}\;\dfrac{G(x)-G(1)}{x-1}}$

$\textbf{Solution:}$

$\mathsf{Consider,}$

$\mathsf{\displaystyle\lim_{x\to\;1}\;\dfrac{G(x)-G(1)}{x-1}}$

$\mathsf{=\displaystyle\lim_{x\to\;1}\;\dfrac{(25-x^2)-(25-1)}{x-1}}$

$\mathsf{=\displaystyle\lim_{x\to\;1}\;\dfrac{-x^2+1}{x-1}}$

$\mathsf{=\displaystyle\lim_{x\to\;1}\;\dfrac{-(x^2-1^2)}{x-1}}$

$\mathsf{=-\displaystyle\lim_{x\to\;1}\;\dfrac{x^2-1^2}{x-1}}$

$\mathsf{Using,}$

$\boxed{\mathsf{\displaystyle\lim_{x\to\;a}\;\dfrac{x^n-a^n}{x-a}=n\,a^{n-1}}}$

$\mathsf{=-2(1)^{2-1}}$

$\mathsf{=-2}$

$\implies\boxed{\mathsf{\displaystyle\lim_{x\to\;1}\;\dfrac{G(x)-G(1)}{x-1}=-2}}$

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