Question:
$f(x)=\frac{|x-a|}{x-a}$ , when $x\neq a=1$ , when $x = a$ then
$f(x)=\frac{|x-a|}{x-a}$ , when $x\neq a=1$ , when $x = a$ then
Updated On: Jul 6, 2022
- $f$ is continuous everywhere
- $f$ is continuous at $x = a$.
- $f$ has a limit 1 at $x = a$
- limit of $f$ does not exist at $x = a$.
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The Correct Option is D
Solution and Explanation
$\lim_{x\to a-} f\left(x\right) = - 1 ,\lim_{x\to a+} f\left(x\right) = 1$
$\therefore$ $\lim_{x\to a} f\left(x\right)$ does not exist.
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.