1. Tardigrade
  2. Question
  3. Mathematics
  4. If G(x) = - √25 - x2 then displaystyle limx → 1 (G(x) - G(1)/x - 1) has the value

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $G(x) = - \sqrt{25 - x^2}$ then $\displaystyle \lim_{x \to 1} \frac{G(x) - G(1)}{x - 1}$ has the value

IIT JEEIIT JEE 1983Limits and Derivatives

Solution:

$\displaystyle\lim_{x \to1} \frac{- \sqrt{25-x^{2}} - \left(-\sqrt{24}\right)}{x-1}$
$ = \displaystyle\lim_{x \to1} \frac{\sqrt{24} - \sqrt{25-x^{2}}}{x-1} \times\frac{\sqrt{24} + \sqrt{25-x^{2}}}{\sqrt{24} + \sqrt{25-x^{2}}} $
$= \displaystyle\lim_{x \to1} \frac{x^{2} - 1}{\left(x-1\right) \left[\sqrt{24} + \sqrt{25 -x^{2}}\right]}$
$ = \displaystyle\lim_{x \to1} \frac{x+1}{ \left[\sqrt{24} + \sqrt{25-x^{2}}\right]} $
$= \frac{2}{2\sqrt{24}} = \frac{1}{2\sqrt{6}} $