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

Problem

(lim_x→0)(x2)*sin(1/x)

Solution

1. Identify the bounds of the sine function, noting that for any real value of θ the value of sin(θ) is always between −1 and 1

|sin(1/x)|≤1

2. Apply the inequality to the entire expression by multiplying all parts by x2 which is always non-negative for x≠0

−x2≤x2*sin(1/x)≤x2

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

(lim_x→0)(−)*x2=0

(lim_x→0)(x2)=0

4. Conclude using the Squeeze Theorem, which states that if the lower and upper bounds approach the same limit, the function trapped between them must also approach that limit.

(lim_x→0)(x2)*sin(1/x)=0

Final Answer

(lim_x→0)(x2)*sin(1/x)=0

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