Question

$\lim_{x \to 0} \frac{\sqrt{ 1 + 2x - 1}}{x} = $

• A. 0
• B. -1
• C. $\frac{1}{2}$
• D. 1
• E. $\frac{-1}{2}$

Answer 1

Answer: (D)

$\displaystyle\lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-1}{x}$
Using L' Hospital's Rule,
$\displaystyle\lim _{x \rightarrow 0} \frac{\frac{1}{2 \sqrt{1+2 x}} \cdot 2-0}{1}$
$\Rightarrow \displaystyle\lim _{x \rightarrow 0} \frac{1}{\sqrt{1+2 x}}$
Using limit, we get
$=\frac{1}{\sqrt{1+2(0)}}=1$