<s>[INST] Evaluate $\lim_{x\to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x-2}$ 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?
 [/INST] Yes it is correct, note also that by definition of derivative with $f(x)=\frac1x \implies f'(x)=-\frac1{x^2}$
$$\lim_{x\to 2} \frac{\frac{1}{x} - \frac{1}{2}}{x-2}=f'(2)=-\frac14$$
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