If $G(x) = - \sqrt{25 - x^2}$ then $\displaystyle \lim_{x \to 1} \frac{G(x) - G(1)}{x - 1}$ has the value
- $\frac{1}{24}$
- $\frac{1}{5}$
- $ - \sqrt{24}$
- none of these
The Correct Option is D
Solution and Explanation
$ = \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}} $
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Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
Properties of Derivatives:
Read More: Limits and Derivatives