4. Applications of Derivatives
Differentials
Problem 4.7.83
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (x + cos x)¹/ˣ
Verified step by step guidance1
Identify the limit to evaluate: lim_{x→0} (x + cos x)^{1/x}. As x approaches 0, both the base (x + cos x) and the exponent (1/x) need to be analyzed.
Evaluate the base as x approaches 0: cos(0) = 1, so x + cos x approaches 0 + 1 = 1.
Recognize that the limit takes the form 1^{∞} as x approaches 0, which is an indeterminate form. To resolve this, take the natural logarithm of the expression: let y = (x + cos x)^{1/x}, then ln(y) = (1/x) * ln(x + cos x).
Now, evaluate the limit of ln(y) as x approaches 0: lim_{x→0} (1/x) * ln(x + cos x). This limit is still indeterminate, so apply l'Hôpital's Rule by differentiating the numerator and denominator.
After applying l'Hôpital's Rule, simplify the resulting expression and evaluate the limit again to find the value of ln(y), then exponentiate to find y.
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