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LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES4 Marks
Question
If the function f(x) satisfies $\lim\limits_{\text{x}\rightarrow1}\frac{\text{f}(\text{x})-2}{\text{x}^2-1}=\pi$ , evaluate $\lim\limits_{\text{x}\rightarrow1}{\text{f}(\text{x})}$.
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Answer

$\lim\limits_{\text{x}\rightarrow1}\frac{\text{f}(\text{x})-2}{\text{x}^2-1}=\pi$$\Rightarrow\frac{\lim\limits_{\text{x}\rightarrow1}{(\text{f}(\text{x})-2)}}{\lim\limits_{\text{x}\rightarrow1}(\text{x}^2-1)}=\pi$
$\Rightarrow{\lim\limits_{\text{x}\rightarrow1}{(\text{f}(\text{x})-2)}}=\pi{\lim\limits_{\text{x}\rightarrow1}(\text{x}^2-1)}$
$\Rightarrow{\lim\limits_{\text{x}\rightarrow1}{(\text{f}(\text{x})-2)}}=0$
$\Rightarrow{\lim\limits_{\text{x}\rightarrow1}{\text{f}(\text{x})}}-{\lim\limits_{\text{x}\rightarrow1}2}=0$
$\Rightarrow{\lim\limits_{\text{x}\rightarrow1}{\text{f}(\text{x})-2}}=0$
$\therefore{\lim\limits_{\text{x}\rightarrow1}{\text{f}(\text{x})}}=2$

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