Evaluate the Limit limit as x approaches 0 of xcos(1/x)

Problem

(lim_x→0)(x)*cos(1/x)

Solution

1. Identify the bounds of the trigonometric function. The cosine function, regardless of its argument, is always bounded between −1 and 1

|cos(1/x)|≤1

2. Apply the inequality to the entire expression. Multiply all parts of the inequality −1≤cos(1/x)≤1 by |x|

−|x|≤x*cos(1/x)≤|x|

3. Evaluate the limits of the bounding functions as x approaches 0

(lim_x→0)(−)*|x|=0

(lim_x→0)(x)=0

4. Apply the Squeeze Theorem. Since the limits of the lower and upper bounds are both 0 the limit of the function trapped between them must also be 0

(lim_x→0)(x)*cos(1/x)=0

Final Answer

(lim_x→0)(x)*cos(1/x)=0

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