Find the limit of the following log terms

Question

I thouht about it alot but could not able to getany idea .

Can anybody provide me a hint

Answer 1

$$\ln(a+x)=\ln a+\ln\left(1+\dfrac xa\right)$$

Now $$\lim_{x\to0}\dfrac{\ln\left(1+\dfrac xa\right)}x=\dfrac1a\lim_{x\to0}\dfrac{\ln\left(1+\dfrac xa\right)}{\dfrac xa}=\dfrac1a$$

Answer 2

Hint:

$$\lim_{x \to 0} \frac{f(3+x) - f(3-x)}{x} = \lim_{x \to 0} \frac{f(3+x) - f(3)}{x} + \lim_{x \to 0} \frac{f(3-x) - f(3)}{-x} = f'(3) + f'(3)$$