4. Applications of Derivatives
Differentials
Problem 4.7.29
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (3 sin 4x) / 5x
Verified step by step guidance1
Identify the limit to evaluate: lim_{x→0} (3 sin(4x)) / (5x).
Substitute x = 0 into the limit to check if it results in an indeterminate form. You will find that both the numerator and denominator approach 0.
Since the limit results in the indeterminate form 0/0, apply l'Hôpital's Rule, which states that you can take the derivative of the numerator and the derivative of the denominator.
Differentiate the numerator: the derivative of 3 sin(4x) is 12 cos(4x), and differentiate the denominator: the derivative of 5x is 5.
Rewrite the limit using the derivatives: lim_{x→0} (12 cos(4x)) / 5, and then substitute x = 0 to evaluate the limit.
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