Calculate the limit: lim (x -> 0) (x^2 sin(1/x))

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Question: Calculate the limit: lim (x -> 0) (x^2 sin(1/x))

Options:

Correct Answer: 0

Solution:

Since |sin(1/x)| ≤ 1, we have |x^2 sin(1/x)| ≤ |x^2|. Thus, lim (x -> 0) x^2 sin(1/x) = 0.

Calculate the limit: lim (x -> 0) (x^2 sin(1/x))

Calculate the limit: lim (x -> 0) (x^2 sin(1/x))

Step 1: Understand the limit we want to calculate: lim (x -> 0) (x^2 sin(1/x)).
Step 2: Recognize that sin(1/x) oscillates between -1 and 1 for all x ≠ 0.
Step 3: This means that the absolute value of sin(1/x) is always less than or equal to 1: |sin(1/x)| ≤ 1.
Step 4: Multiply both sides of the inequality by x^2 (which is always non-negative): |x^2 sin(1/x)| ≤ |x^2|.
Step 5: Now we need to find the limit of |x^2| as x approaches 0: lim (x -> 0) |x^2| = 0.
Step 6: Since |x^2 sin(1/x)| is squeezed between -|x^2| and |x^2|, we can apply the Squeeze Theorem.
Step 7: By the Squeeze Theorem, since lim (x -> 0) |x^2| = 0, we conclude that lim (x -> 0) x^2 sin(1/x) = 0.

Limit of a Function – Understanding how to evaluate the limit of a function as the variable approaches a specific value.
Squeeze Theorem – Applying the Squeeze Theorem to find limits of functions that oscillate, such as sin(1/x).
Behavior of Trigonometric Functions – Recognizing the bounded nature of the sine function and its implications for limits.