4. Applications of Derivatives
Differentials
Problem 4.7.20
Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - 1) / (2x + 5)
Verified step by step guidance1
Identify the limit to evaluate: lim_{x→0} (e^x - 1) / (2x + 5).
Substitute x = 0 into the expression to check if it results in an indeterminate form.
Since substituting x = 0 gives (e^0 - 1) / (2(0) + 5) = (1 - 1) / 5 = 0 / 5 = 0, we can proceed with l'Hôpital's Rule.
Differentiate the numerator and the denominator separately: the derivative of e^x is e^x, and the derivative of (2x + 5) is 2.
Rewrite the limit using the derivatives: lim_{x→0} (e^x) / (2) and then substitute x = 0 to find the limit.
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