Examine the following functions for continuity. f(x) = 1 x-5 - Vidyadip

Question

Examine the following functions for continuity.
$f(x) = \frac{1}{\text{x}-5}$

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Answer

Here $f(x) = \frac{1}{\text{x}-5}$ For $f$ to be defined,
$x - 5 \neq 0$ i.e. x $\neq 5$
$\therefore D_f=$ Set of real number except $5 = R = -{5}$
Let $c \neq 5$ be any real number.
Also $^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)}-^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\Big(\frac{1}{\text x- 5}\Big)= \frac{1}{\text{c}-5}$
$\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = \text{f(c)}$
$\therefore$ f is continuous at $x = c.$
But c $\neq 5$ is any real number
$\therefore f$ is continuous at every real number $\text{c}\in \text{D}$
$\therefore f$ is continuous function.

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