To show: The limit of the function .
Theorem used: The Squeeze Theorem
“If when x is near a (except possibly at a) and then .”
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 .
Let , and .
When the limit x approaches zero, the inequality becomes,
Sketch the graph of the function by using the online graphing calculator as shown below in Figure (1).
From the graph, it is observed that and .
If when x approaches 0 and , then by Squeeze Theorem the limit of the function is zero.
That is, .
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