The value of $\displaystyle\lim_{x\to0} \frac{\left|x\right|}{x} $ is
- 1
- -1
- 0
- Does not exist
The Correct Option is D
Solution and Explanation
$RHL =\displaystyle\lim _{ x \rightarrow 0} f (0+ h )=\displaystyle\lim _{ h \rightarrow 0} \frac{|0+ h |}{0+ h }=\displaystyle\lim _{ h \rightarrow 0} \frac{ h }{ h }=\displaystyle\lim _{ h \rightarrow 0}(1)=1$
$\therefore \,LHL \neq RHL$
$ \Rightarrow \,\displaystyle\lim _{x \rightarrow 0} \frac{| x |}{ x }$ does not exist
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).