Question

if f(r)=pi×r^2 then find lim h-0[f(r+h)-f(r)/h]​

Answer 1

Topic - Limits

Given. $f(r)=\pi r^{2}$

To find. $\displaystyle \lim_{h\to 0}\frac{f(r+h)-f(r)}{h}$

Solution.

Here, $f(r)=\pi r^{2}$

Then, $f(r+h)=\pi (r+h)^{2}$

Now, $\displaystyle \lim_{h\to 0}\frac{f(r+h)-f(r)}{h}$

$\displaystyle =\lim_{h\to 0}\frac{\pi (r+h)^{2}-\pi r^{2}}{h}$

$\displaystyle =\lim_{h\to 0}\frac{\pi \{(r+h)^{2}-r^{2}\}}{h}$

$\displaystyle =\lim_{h\to 0}\frac{\pi (r+h+r)(r+h-r)}{h}$

$\displaystyle =\lim_{h\to 0}\frac{\pi (2r+h) h}{h}$

$\displaystyle =\lim_{h\to 0}\pi (2r+h)$ since $h\neq 0$

$=\pi(2r+0)$

$=2\pi r$

Answer. $\displaystyle \lim_{h\to 0}\frac{f(r+h)-f(r)}{h}=2\pi r$

Note. The given limit presents the derivative of $f(r)$ with respect to $r$. So this can be written as $\frac{d}{dr}(\pi r^{2})=2\pi r$.