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Question: Find the limit: \[\lim_ {x \to 0}\] for \[\sin \dfrac{\left| x \right|}{x}\]?...

Find the limit: lim⁡x→0\lim_ {x \to 0} for sin⁡∣x∣x\sin \dfrac{\left| x \right|}{x}?

Explanation

Solution

We are given to find the limit of \lim_{x \to 0}$$$$\sin \dfrac{\left| x \right|}{x}, in order to find the limit firstly, we will be computing the limit considering x<0x<0. And then we will be computing the limit considering x>0x>0. After finding both of the limits then we will be determining the limit if it exists.

Complete step-by-step solution:
Now let us have a brief regarding finding the limit of functions. In order to find the limit, we have to find the LCDLCD of fractions on the top. And then we have to solve it by distributing the numerators and then subtracting or adding the numerators and then we have to solve them by applying the fraction rules to it. At last we are supposed to substitute the value into the obtained function and then evaluate it.
Now let us find the limit of lim⁡x→0sin⁡∣x∣x\lim_{x \to 0} \sin \dfrac{\left| x \right|}{x}.
We will be finding the limits as the xx approaches from the left as well as from the right.
Now consider for x<0x<0
⇒∣x∣=−x\Rightarrow \left| x \right|=-x
Now let us compute the function. We get
lim⁡x→0−sin⁡∣x∣x\displaystyle \lim_{x \to 0}-\sin \dfrac{\left| x \right|}{x}
\Rightarrow $$$$\displaystyle \lim_{x \to 0}-\sin \dfrac{\left( -x \right)}{x}
\Rightarrow $$$$-\displaystyle \lim_{x \to 0} \sin \dfrac{\left( x \right)}{x}
=−1=-1
Now let us consider for x>0x>0
⇒∣x∣=x\Rightarrow \left| x \right|=x
Upon computing it, we get
⇒lim⁡x→0+sin⁡∣x∣x\Rightarrow \displaystyle \lim_{x \to 0}+\sin \dfrac{\left| x \right|}{x}
⇒lim⁡x→0+sin⁡xx\Rightarrow \displaystyle \lim_{x \to 0}+\sin \dfrac{x}{x}
=1=1
We can observe that the limits from right as well as from the left are different. Therefore, we conclude that no limit exists for the given function.
∴\therefore lim⁡x→0sin⁡∣x∣x\displaystyle \lim_{x \to 0}\sin \dfrac{\left| x \right|}{x} does not exist.

Note: We must note that the limit of a constant times a function is equal to the constant times the limit of a function. When the limits are different, the limit does not exist. We can also compute the limits by using the L-Hospital’s rule.