Question-12

If \(\displaystyle \lim\limits _{x\rightarrow 4} \ \frac{f( x) -5}{x-2} =1\), find \(\displaystyle \lim\limits _{x\rightarrow 4} \ f( x)\).

Hint

Use the property of limits (arithmetic).

Answer

\(7\)

Solution

Using the multiplicative property of limits:

\[ \begin{equation*} \begin{aligned} \lim\limits _{x\rightarrow 4} \ f( x) -5 & =\lim\limits _{x\rightarrow 4} \ \frac{f( x) -5}{x-2} \cdot ( x-2)\\ & =\lim\limits _{x\rightarrow 4} \ \frac{f( x) -5}{x-2} \cdot \lim\limits _{x\rightarrow 4} \ x-2\\ & =1\times 2\\ & =2 \end{aligned} \end{equation*} \]

Therefore, \(\displaystyle \lim\limits _{x\rightarrow 4} \ f( x) =7\).