Evaluate the Limit limit as x approaches 0 of x^2cos(1/x)

Problem

(lim_x→0)(x^2)*cos(1/x)

Solution

1. Identify the bounds of the trigonometric function. The cosine function, cos(1/x) is bounded between −1 and 1 for all x≠0

−1≤cos(1/x)≤1

2. Apply the squeeze theorem by multiplying the inequality by x^2 Since x^2≥0 for all real x the direction of the inequalities remains the same.

−x^2≤x^2*cos(1/x)≤x^2

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

(lim_x→0)(−)*x^2=0

(lim_x→0)(x^2)=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→0)(x^2)*cos(1/x)=0

Final Answer

(lim_x→0)(x^2)*cos(1/x)=0

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