Question

Evaliate lim x--->pi/2 cotx.log (secx) By I Hospital Rule​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: cotx. \: log(secx)$

can be rewritten as

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \frac{log(secx)}{tanx}$

If we substitute directly the value of x, we get

$\rm \: = \: \dfrac{ \infty }{ \infty }$

which is indeterminant form.

So, Consider again

$\rm \: \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \frac{log(secx)}{tanx}$

On applying L Hospital Rule, we get

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \frac{\dfrac{d}{dx}log(secx)}{\dfrac{d}{dx}tanx}$

We know,

$\boxed{ \rm{ \:\dfrac{d}{dx}logx \: = \: \frac{1}{x} \: }}$

and

$\boxed{ \rm{ \:\dfrac{d}{dx}tanx \: = \: {sec}^{2}x \: }}$

So, using these results, we get

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{\dfrac{1}{secx}\dfrac{d}{dx}(secx) }{ {sec}^{2} x}$

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{secx \: tanx }{ {sec}^{3} x}$

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{ tanx }{ {sec}^{2} x}$

On applying L Hospital Rule, we get

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{\dfrac{d}{dx} tanx }{\dfrac{d}{dx} {sec}^{2} x}$

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{{sec}^{2}x}{2 \: secx \: (secx \: tanx)}$

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{{sec}^{2}x}{2 \: {sec}^{2} x \: tanx}$

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{1}{2 tanx}$

$\rm \: = \: \displaystyle\lim_{x \to \dfrac{\pi}{2} } \: \dfrac{cotx}{2}$

$\rm \: = \: \dfrac{1}{2} \times 0$

$\rm \: = \: 0$

Hence,

$\\ \color{green}\rm\implies \:\boxed{ \rm{ \:\displaystyle\lim_{x \to \dfrac{\pi}{2} } \: cotx. \: log(secx) = 0 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {x}^{n}\\ \\ \sf {nx}^{n - 1} & \sf {e}^{x} \end{array}} \\ \end{gathered}$