4. Applications of Derivatives
Differentials
Problem 4.7.75
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0⁺ x²ˣ
Verified step by step guidance1
Recognize that the limit lim_{x→0⁺} x^{2^x} can be rewritten using the exponential function: x^{2^x} = e^{2^x imes ext{ln}(x)}.
Identify that as x approaches 0 from the right, ln(x) approaches -∞, and 2^x approaches 1, leading to the form 1 × -∞, which is indeterminate.
To resolve the indeterminate form, rewrite the limit as lim_{x→0⁺} 2^x imes ext{ln}(x) and apply l'Hôpital's Rule if necessary, focusing on the limit of ln(x) as x approaches 0.
Differentiate the numerator and denominator if using l'Hôpital's Rule, or consider the behavior of the function as x approaches 0 to simplify the limit.
Evaluate the resulting limit to find the final value, ensuring to interpret the behavior of the exponential function correctly.
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