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Calculus: Evaluating Limits with Indeterminate Products

  • Nov 4, 2024
  • 1 min read

Updated: 3 days ago

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Evaluate the following limit:

Mathematical expression on dark background: limit as x approaches infinity of x^2 sin(1/4x^2). White handwritten text.

This is an example of a limit with indeterminate form ∞ ⋅ 0 since:

White handwritten math equations on a dark background, showing limits: lim x→∞ [x²] = ∞ and lim x→∞ [sin(1/4x²)] = 0.

This is called an indeterminate product.

To avoid a difficult solution involving L'Hôpital's Rule, we can make the following observations:

Observation 1:

As x → ∞, 1/4x² → 0

Observation 2:

As x → 0, sin(x) → x

Therefore: As x → ∞, sin(1/4x²) → 1/4x²

Now we can finish the limit as follows:

Math equation on a black background showing a limit calculation where the limit of x² sin(1/4x²) as x approaches infinity equals 1/4.

Final answer:

x = 1/4

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