Question: What is the value of the limit lim (x -> 0) (sin(5x)/x)?
Options:
0
1
5
10
Correct Answer: 5
Solution:
Using L\'Hôpital\'s rule, the limit evaluates to 5.
What is the value of the limit lim (x -> 0) (sin(5x)/x)?
Practice Questions
Q1
What is the value of the limit lim (x -> 0) (sin(5x)/x)?
0
1
5
10
Questions & Step-by-Step Solutions
What is the value of 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: Notice that if we plug in x = 0 directly, we get sin(0)/0, which is 0/0. This is an indeterminate form.
Step 3: Since we have an indeterminate form, we can use L'Hôpital's rule. This rule states that if we have 0/0 or ∞/∞, we can take the derivative of the top and the derivative of the bottom.
Step 4: Differentiate the numerator: The derivative of sin(5x) is 5cos(5x).
Step 5: Differentiate the denominator: The derivative of x is 1.
Step 6: Now we rewrite the limit using these derivatives: lim (x -> 0) (5cos(5x)/1).
Step 7: Plug in x = 0 into the new limit: 5cos(5*0) = 5cos(0) = 5*1 = 5.
Step 8: Therefore, the value of the limit is 5.
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