Examine the function for continuity. f(x)=1 x-5 , x 5 - Vidyadip

Question

Examine the function for continuity.
$f(x)=\frac{1}{x-5}$, x $\neq$ 5

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Answer

The given function is $f(x)=\frac{1}{x-5}, x \neq 5$
For any real number $k \neq 5$ we get,
$\lim _{x \rightarrow k} f(x)=\lim _{x \rightarrow k} \frac{1}{(x-5)}=\frac{1}{(k-5)}$
Also, $f(k)=\frac{1}{(k-5)}(As, ~~ k \neq 5)$
Thus, $\mathop {\lim }\limits_{{{x}} \to {{k}}} {{f}}({{x}}) = {{f}}({{k}})$
Therefore, f is continuous at every point in the domain of f and thus, it is a continuous function.

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