Question

What is $\underset{h\to 0}{lim}\frac{\sqrt{2x+3h}-\sqrt{2x}}{2h}$ equal to?

• A. $\frac{1}{2\sqrt{2x}}$
• B. $\frac{3}{\sqrt{2x}}$
• C. $\frac{3}{2\sqrt{2x}}$
• D. $\frac{3}{4\sqrt{2x}}$

Answer 1

Answer: (D)

$\frac{3}{4\sqrt{2x}}$

${lim}_{h\to 0}\frac{\sqrt{2x+3h}-\sqrt{2x}}{2h}=\frac{0}{0}$

As we have $\frac{0}{0}$ form, we can use L'Hospital's rule wrt $h$.

${lim}_{h\to 0}\frac{\sqrt{2x+3h}-\sqrt{2x}}{2h}={lim}_{h\to 0}\frac{\frac{1}{2\sqrt{2x+3h}}\times 3}{2}=\frac{3}{4\sqrt{2x}}$