Evaluate the Limit limit as x approaches 0 of x^4cos(2/x)

Problem

(lim_x→0)(x4)*cos(2/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(2/x)|≤1

2. Set up an inequality for the entire expression. Since x4 is always non-negative for any real x multiply the inequality −1≤cos(2/x)≤1 by x4

−x4≤x4*cos(2/x)≤x4

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

(lim_x→0)(−)*x4=0

(lim_x→0)(x4)=0

4. Apply the Squeeze Theorem. Since the limits of the outer functions are both equal to 0 the limit of the function squeezed between them must also be 0

(lim_x→0)(x4)*cos(2/x)=0

Final Answer

(lim_x→0)(x4)*cos(2/x)=0

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