Question

lim x_4 {1/log (x-3) - 1/x-4 }

Answer 1

Step-by-step explanation:

$\sf \red{\underset{x \rightarrow 4}{lim}\left\{\frac{1}{\log (x-3)}-\frac{1}{x-4}\right\}}$

$\sf \pink{\underset{X \rightarrow 4}{ lim}\left \{\dfrac{x-4-\ln (x-3)}{\ln (x-3)(x-4)}\right \}}$

$\sf \purple{\underset{X \rightarrow 4}{lim}\left \{\dfrac{-\frac{1}{x-3}+1}{\frac{x-4}{x-3}+\ln (x-3)}\right \} }$

$\sf \blue{\underset{X \rightarrow 4}{lim}\left \{\dfrac{X-4}{X-4+\ln (X-3)(X-3)}\right \}}$

Apply L'Hopital's Rule

$\sf \green{\underset{X \rightarrow 4}{Lim}\left \{\frac{1}{\ln (X-3)+2}\right \}}$

Plug In The Value X=4,We Get,

$\tt \orange{ = \dfrac{1}{ \ln(4 - 3) + 2} }$

$\tt \purple{ = \dfrac{1}{2} }$