If the function f(x) satisfies
Question:

If the function $1(x)$ satisfies $\lim _{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi$, evaluate $\lim _{x \rightarrow 1} f(x)$.

Solution:

$\lim _{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi$

$\Rightarrow \frac{\lim _{x \rightarrow 1}(f(x)-2)}{\lim _{x \rightarrow 1}\left(x^{2}-1\right)}=\pi$

$\Rightarrow \lim _{x \rightarrow 1}(f(x)-2)=\pi \lim _{x \rightarrow 1}\left(x^{2}-1\right)$

$\Rightarrow \lim _{x \rightarrow 1}(f(x)-2)=\pi\left(1^{2}-1\right)$

$\Rightarrow \lim _{x \rightarrow 1}(f(x)-2)=0$

$\Rightarrow \lim _{x \rightarrow 1} f(x)-\lim _{x \rightarrow 1} 2=0$

$\Rightarrow \lim _{x \rightarrow 1} f(x)-2=0$

$\therefore \lim _{x \rightarrow 1} f(x)=2$

 

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