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Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability1 Mark
Question
Determine the value of $k$ for which the function $f(x)$ is continuous at $x=4$.
$f(x)=\left\{\begin{array}{ll} \frac{x^2-16}{x-4}, & x \neq 4 \\ k, & x=4 \end{array}\right. $
✓

Answer

$(d):$ Since $f(x)$ is continuous at $x=4$. Therefore,
$\lim _{x \rightarrow 4} f(x)=f(4)$
$\Rightarrow \lim _{x \rightarrow 4} f(x)=k \quad[\because f(4)=k]$
$\Rightarrow \lim _{x \rightarrow 4} \frac{x^2-16}{x-4}=k \Rightarrow \lim _{x \rightarrow 4} \frac{(x-4)(x+4)}{x-4}=k$
$\Rightarrow \lim _{x \rightarrow 4}(x+4)=k \Rightarrow k=8$

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