Question

The value of $\lim_{x\rightarrow 0}\frac{\sqrt{1+x} - 1}{x} = $

• A. 0
• B. $1/2$
• C. 1
• D. $\infty$

Hint:

For $\sqrt{1+x}$, the binomial approximation for small $x$ is $1 + \frac{1}{2}x$. Substituting this makes the limit $(1 + \frac{1}{2}x - 1)/x = 1/2$ instantly.

Answer 1

The Correct Option is B

Step 1: Concept
The limit results in an indeterminate form $0/0$. We can use rationalization or L'Hôpital's Rule.

Step 2: Meaning
Differentiate the numerator and denominator separately using L'Hôpital's Rule.

Step 3: Analysis
$\frac{d}{dx}(\sqrt{1+x} - 1) = \frac{1}{2\sqrt{1+x}}$ and $\frac{d}{dx}(x) = 1$. The limit becomes $\lim_{x\rightarrow 0} \frac{1}{2\sqrt{1+x}}$.

Step 4: Conclusion
Substituting $x = 0$ gives $1/(2\sqrt{1}) = 1/2$.
Final Answer: (B)