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Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x

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Question: Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x = 0?

Options:

  1. 5, Continuous
  2. 5, Discontinuous
  3. 0, Continuous
  4. 0, Discontinuous

Correct Answer: 5, Continuous

Solution: 

Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0 if defined as f(0) = 5.

Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x

Practice Questions

Q1
Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x = 0?
  1. 5, Continuous
  2. 5, Discontinuous
  3. 0, Continuous
  4. 0, Discontinuous

Questions & Step-by-Step Solutions

Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x = 0?
  • Step 1: Understand the limit we want to evaluate: lim (x -> 0) (sin(5x)/x).
  • Step 2: Recall 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: To check if the function is continuous at x = 0, we need to define f(0).
  • Step 6: If we define f(0) = 5, then the function is continuous at x = 0.
  • Limit Evaluation – Understanding how to evaluate limits involving trigonometric functions and their properties.
  • Continuity – Determining if a function is continuous at a point by checking if the limit equals the function's value at that point.
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