Finding the limit of $f(x)$

Question

Using the given condition to find$$ \lim_{x \to 2} f(x).$$

Answer 1

Note that since

$$f(x)=f(2)+f'(2)(x-2)+o(x-2)$$

$$\lim_{x\to2}\frac{f(x)-(x+2)}{x-2}=\frac{5}{6}\implies\lim_{x\to2}\frac{f(2)+f'(2)(x-2)-(x+2)+o(x-2)}{x-2}=\frac56$$

the limit exists if and only if

$$\lim_{x\to2} f(2)-(x+2)=0\implies f(2)=4$$

and since $f(x)$ is continuos

$$\lim_{x\to2} f(x)=f(2)=4$$