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

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Question: Find 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|. As x approaches 0, |x^2| approaches 0, hence the limit is 0.

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

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

Step 1: Understand the limit we want to find: lim (x -> 0) (x^2 * sin(1/x)).
Step 2: Recall that the sine function, sin(y), always has values between -1 and 1 for any real number y.
Step 3: Since 1/x becomes very large as x approaches 0, we know that sin(1/x) will oscillate between -1 and 1.
Step 4: This means that |sin(1/x)| is always less than or equal to 1, so we can say |x^2 * sin(1/x)| <= |x^2|.
Step 5: Now, we look at |x^2| as x approaches 0. As x gets closer to 0, x^2 also gets closer to 0.
Step 6: Therefore, since |x^2 * sin(1/x)| is less than or equal to |x^2|, and |x^2| approaches 0, we conclude that |x^2 * sin(1/x)| also approaches 0.
Step 7: Finally, we can say that the limit is 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 are bounded by other functions.
Behavior of Trigonometric Functions – Recognizing the bounded nature of the sine function and its implications for limits.