4. Applications of Derivatives
Differentials
Problem 18
Textbook Question
Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)
Verified step by step guidance1
First, substitute x = -1 into the limit expression to check if it results in an indeterminate form. If it does, we can apply l'Hôpital's Rule.
Calculate the numerator: (-1)⁴ + (-1)³ + 2(-1) + 2, and the denominator: -1 + 1. Confirm if both yield 0.
Since both the numerator and denominator are 0, apply l'Hôpital's Rule by differentiating the numerator and the denominator separately.
Differentiate the numerator: f(x) = x⁴ + x³ + 2x + 2, so f'(x) = 4x³ + 3x² + 2. Differentiate the denominator: g(x) = x + 1, so g'(x) = 1.
Now, evaluate the limit again using the derivatives: lim_x→ -1 (4x³ + 3x² + 2) / 1, and substitute x = -1 to find the limit.
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