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Evaluate the limit: lim (x -> 0) (sin(5x)/x)

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Question: Evaluate the limit: lim (x -> 0) (sin(5x)/x)

Options:

  1. 0
  2. 5
  3. 1
  4. Infinity

Correct Answer: 5

Solution: 

Using the standard limit lim (x -> 0) (sin(x)/x) = 1, we have lim (x -> 0) (sin(5x)/x) = 5 * lim (x -> 0) (sin(5x)/(5x)) = 5 * 1 = 5.

Evaluate the limit: lim (x -> 0) (sin(5x)/x)

Practice Questions

Q1
Evaluate the limit: lim (x -> 0) (sin(5x)/x)
  1. 0
  2. 5
  3. 1
  4. Infinity

Questions & Step-by-Step Solutions

Evaluate the limit: lim (x -> 0) (sin(5x)/x)
  • Step 1: Identify the limit we want to evaluate: lim (x -> 0) (sin(5x)/x).
  • Step 2: Recall the standard limit: lim (x -> 0) (sin(x)/x) = 1.
  • Step 3: Notice that we can rewrite sin(5x) in terms of sin(5x)/(5x).
  • Step 4: Rewrite the limit: lim (x -> 0) (sin(5x)/x) = lim (x -> 0) (sin(5x)/(5x)) * 5.
  • Step 5: Now, we can apply the standard limit: lim (x -> 0) (sin(5x)/(5x)) = 1.
  • Step 6: Multiply the result by 5: 5 * 1 = 5.
  • Step 7: Conclude that the limit is 5.
  • Limit Evaluation – Understanding how to evaluate limits involving trigonometric functions and applying standard limit results.
  • Substitution in Limits – Using substitution to transform the limit into a more manageable form.
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