Question

$\underset{x\to 0}{lim}\frac{{(x+2)}^{10}-2^{10}}{{(x+2)}^5-2^5}$ is ______

• A. 16
• B. 32
• C. 64
• D. 128

Answer 1

Answer: (C)

64

Given,
$\underset{x\to 0}{lim}\frac{{(x+2)}^{10}-2^{10}}{{(x+2)}^5-2^5}=\underset{x\to 0}{lim}\frac{{\left(\right(x+2{)}^5)}^2-{\left(2^5\right)}^2}{{(x+2)}^5-2^5}$
$=\underset{x\to 0}{lim}\frac{{\left(\right(x+2)}^5-2^5\left)\right({(x+2)}^5+2^5)}{{(x+2)}^5-2^5}$ Since $a^2-b^2=(a+b)(a-b)$
$=\underset{x\to 0}{lim}({(x+2)}^5+2^5)$
Substituting $x=0$, we get
$={(x+2)}^5+2^5=2^5+2^5=32+32=64$