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Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)

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Question: Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)

Options:

  1. 5, Continuous
  2. 0, Not continuous
  3. 5, Not continuous
  4. 0, Continuous

Correct Answer: 5, Continuous

Exam Year: 2021

Solution: 

Using the limit property, lim (x -> 0) (sin(kx)/x) = k. Here, k = 5, so the limit is 5, and the function is continuous at x = 0.

Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)

Practice Questions

Q1
Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)
  1. 5, Continuous
  2. 0, Not continuous
  3. 5, Not continuous
  4. 0, Continuous

Questions & Step-by-Step Solutions

Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)
  • Step 1: Identify the limit we need to evaluate: lim (x -> 0) (sin(5x)/x).
  • Step 2: Recognize the limit property: lim (x -> 0) (sin(kx)/x) = k, where k is a constant.
  • Step 3: In our case, k = 5 because we have sin(5x).
  • Step 4: Apply the limit property: lim (x -> 0) (sin(5x)/x) = 5.
  • Step 5: Conclude that the limit is 5.
  • Step 6: Determine continuity: Since sin(5x)/x is defined and approaches 5 as x approaches 0, the function is continuous at x = 0.
  • Limit Evaluation – Understanding how to evaluate limits involving trigonometric functions, specifically using the property of limits for sin(kx)/x.
  • Continuity – Determining the continuity of a function at a point, particularly at x = 0 in this case.
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