4. Applications of Derivatives
Differentials
Problem 4.7.22
Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - 1) / (x² + 3x)
Verified step by step guidance1
Identify the limit to evaluate: lim_{x→0} (e^x - 1) / (x² + 3x). Check if it results in an indeterminate form by substituting x = 0 into the numerator and denominator.
Substituting x = 0 gives (e^0 - 1) / (0² + 3*0) = (1 - 1) / 0 = 0/0, which is an indeterminate form. This means we can apply l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and the denominator separately. The derivative of the numerator e^x - 1 is e^x, and the derivative of the denominator x² + 3x is 2x + 3.
Rewrite the limit using the derivatives: lim_{x→0} e^x / (2x + 3). Now substitute x = 0 again to evaluate the new limit.
After substituting x = 0, calculate e^0 / (2*0 + 3) = 1 / 3. This is the final value of the limit.
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