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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
Question
Identify discontinuities for the following functions as either a jump or a removable discontinuity : $f(x)=\frac{x^2-10 x+21}{x-7}$
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Answer

Given, $f(x)=\frac{x^2-10 x+21}{x-7}$
It is a rational function and is discontinuous if $x-7=0$,
i.e., $x=7$
$\therefore f ( x )$ is continuous for all $x \in R$, except at $x =7$.
$\therefore f (7)$ is not defined.
Now, $\lim _{x \rightarrow 7} f (x) =\lim _{x \rightarrow 7} \frac{x^2-10 x+21}{x-7}$
$ =\lim _{x \rightarrow 7} \frac{(x-7)(x-3)}{x-7}$
$=\lim _{x \rightarrow 7}(x-3) \ldots\left[\begin{array}{l} \because x \rightarrow 7, x \neq 7, \\ \therefore x-7 \neq 0 \end{array}\right]$
$=7-3$
$ =4$
Thus, $\lim _{x \rightarrow 7} f (x)$ exist but $f (7)$ is not defined.
​​​​​​​$\therefore f ( x )$ has a removable discontinuity.

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