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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