Limit Laws

Problem 146: If $\lim_{x \to 4} \frac{f(x)-5}{x-2} = 1$ , find $\lim_{x \to 4} f(x)$ .

Solution: We will use the limit laws to solve this problem.

(1) $\begin{align*} \lim_{x \to 4} \frac{f(x)-5}{x-2} & = 1 \\ \frac{\lim_{x \to 4} f(x) - 5}{\lim_{x \to 4} x - 2} & = 1 \\ \lim_{x \to 4} f(x) - 5 & = 1 \cdot \left({\lim_{x \to 4} x - 2}\right) \\ \lim_{x \to 4} f(x) & = 1 \cdot \left({\lim_{x \to 4} x - 2}\right) + 5 \\ & = 4-2+5 = 7. \end{align*}$

That is,

(2) $\begin{equation*} \boxed{ \lim_{x \to 4} f(x) = 7}.\end{equation*}$

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