Evaluate the Limit

Problem

(lim_x→∞)(sin(x)/x)

Solution

1. Identify the bounds of the numerator. The function sin(x) oscillates between −1 and 1 for all real values of x

For all *x,−1≤sin(x)≤1

2. Apply the Squeeze Theorem by dividing the entire inequality by x Since we are evaluating the limit as x→∞ we can assume x>0

(−1)/x≤sin(x)/x≤1/x

3. Evaluate the limits of the lower and upper bounding functions as x approaches infinity.

(lim_x→∞)((−1)/x)=0

(lim_x→∞)(1/x)=0

4. Conclude using the Squeeze Theorem. Since both the lower and upper bounds approach 0 the limit of the middle function must also be 0

(lim_x→∞)(sin(x)/x)=0

Final Answer

(lim_x→∞)(sin(x)/x)=0

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