_ x 0 a^ 3 x -a^ 2 x -a^x+1 x^2 is equal to - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{a^{3 x}-a^{2 x}-a^x+1}{x^2}$ is equal to

A
$\sqrt{2} \log a^2$
B
$\sqrt{2} \log a$
✓
$2(\log a)^2$
D
$2(\log a)^3$

✓

Answer

Correct option: C.

$2(\log a)^2$

(C)
$\lim _{x \rightarrow 0} \frac{ a ^{3 x}- a ^{2 x}- a ^x+1}{x^2}$
$=\lim _{x \rightarrow 0} \frac{ a ^{2 x} \cdot a ^x- a ^{2 x}- a ^x+1}{x^2}$
$=\lim _{x \rightarrow 0} \frac{\left( a ^{2 x}-1\right)\left( a ^x-1\right)}{x^2}$
$=2 \log a \cdot \log a =2(\log a )^2$

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