Question

Evaluate:

$\underset{x\to 2}{lim}\frac{x^{\frac{5}{2}}-2^{\frac{5}{2}}}{x^{\frac{7}{2}}-2^{\frac{7}{2}}}$.

Answer 1

Now,

$\underset{x\to 2}{lim}\frac{x^{\frac{5}{2}}-2^{\frac{5}{2}}}{x^{\frac{7}{2}}-2^{\frac{7}{2}}}$ $\left(\frac{0}{0}\right)$ form.

Now applying L'Hospital's rule we get,

$=\underset{x\to 2}{lim}\frac{\frac{5}{2}\times x^{\frac{3}{2}}}{\frac{7}{2}\times x^{\frac{5}{2}}}$

$=\frac{5}{7}\times \frac{1}{2}$

$=\frac{5}{14}$.