To prove: The limit of the function .
Theorem used: The Squeeze Theorem
“If when x is near a (except possibly at a) and then .”
It is trivial that, the value of does not exist.
Thus, the limit of the function does not exist.
Apply the Squeeze Theorem and obtain a function f smaller than and a function h bigger than such that both and approaches 0.
Since the cosine function is lies between and 1, .
Any inequality remains true when multiplied by a positive number. Since for all x, multiply each side of the inequalities by .
When the limit x approaches zero, the inequality becomes,
Use the online graphing calculator to draw the graph of the function as shown below in Figure 1.
From Figure 1, it is observed that and .
Let , and .
If when x approaches 0 and , then by Squeeze Theorem the limit of the function is zero.
That is, .
Hence the required proof is obtained.
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