Lim X → 0 a X + a − X − 2 X 2 - Mathematics

$\lim_{x \to 0} \frac{a^x + a^{- x} - 2}{x^2}$

Solution

$\lim_{x \to 0} \left[ \frac{a^x + a^{- x} - 2}{x^2} \right]$
$= \lim_{x \to 0} \left[ \frac{\left( a^\frac{x}{2} \right)^2 + \left( a^{- \frac{x}{2}} \right)^2 - 2 a^\frac{x}{2} \cdot a^{- \frac{x}{2}}}{x^2} \right]$
$= \lim_{x \to 0} \left[ \frac{\left( a^\frac{x}{2} - a^{- \frac{x}{2}} \right)^2}{x^2} \right]$
$= \lim_{x \to 0} \left[ \frac{\left( a^\frac{x}{2} - \frac{1}{a^\frac{x}{2}} \right)^2}{x^2} \right]$
$= \lim_{x \to 0} \left[ \frac{\left( a^x - 1 \right)^2}{x^2} \times \frac{1}{\left( a^\frac{x}{2} \right)^2} \right]$
$= \lim_{x \to 0} \left[ \left( \frac{a^x - 1}{x} \right)^2 \times \frac{1}{a^x} \right]$
$= \frac{\left( \log a \right)^2}{a^0}$
$= \left( \log a \right)^2$

Is there an error in this question or solution?

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.1 | Q 3 | Page 71