Evaluate $\lim_{x\to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x-2}$

Question

Evaluate:

$$\lim_{x\to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x-2}$$

My attempt: $$\lim _{ x\rightarrow 2 }{ \frac { \frac 1 x -\frac 1 2 }{ x-2 } } =\lim_{ x\rightarrow 2 }{ \frac { \frac { 2-x }{ 2x } }{ x-2 } } =-\lim_{ x\rightarrow 2 }{ \frac 1 {2x} =-\frac 1 4} $$

Is this correct?

Answer 1

Yes. Your answer is correct. You can also do it by using L'Hospitals rule.

$$\begin{align}\lim_{x \to 2}\dfrac{\dfrac{1}{x}-\dfrac{1}{2}}{x-2}&=\lim_{x\to 2}\left(-\dfrac{1}{x^2}\right) \\ &=-\dfrac{1}{4}\end{align}\tag*{}$$

Answer 2

Yes it is correct, note also that by definition of derivative with $f(x)=\frac1x \implies f'(x)=-\frac1{x^2}$

$$\lim_{x\to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x-2}=f'(2)=-\frac14$$

Answer 3

"Yes" would suffice as an answer.