Indeterminate Powers and Products
Find the limits in Exercises 53–68.
60. lim (x → 0) (e^x + x)^(1/x)

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Identify the limit expression: \(\lim_{x \to 0} \left(e^{x} + x\right)^{\frac{1}{x}}\).

Recognize that the expression is of the form \(f(x)^{g(x)}\) where both the base and the exponent approach values that create an indeterminate form. To handle this, rewrite the limit using the exponential and natural logarithm functions: \(\lim_{x \to 0} \exp\left( \frac{1}{x} \cdot \ln\left(e^{x} + x\right) \right)\).

Focus on the inner limit: \(\lim_{x \to 0} \frac{\ln\left(e^{x} + x\right)}{x}\). This is a \(\frac{0}{0}\) indeterminate form, so consider applying L'Hôpital's Rule or use series expansions to simplify the numerator and denominator.

Use the Taylor series expansions around \(x=0\) for \(e^{x}\) and \(\ln(1 + y)\) to approximate \(e^{x} + x\) and then \(\ln(e^{x} + x)\), which will help simplify the expression inside the limit.

After simplifying the inner limit, substitute back into the exponential function to find the overall limit.

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Limits Involving Indeterminate Forms

When evaluating limits, expressions may take forms like 0^0, ∞^0, or 1^∞, which are indeterminate and require special techniques to resolve. Recognizing these forms is crucial to apply appropriate methods such as logarithmic transformation or L'Hôpital's Rule.

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Integrals Involving Natural Logs: Substitution

Logarithmic Transformation for Limits

Transforming a limit of the form f(x)^g(x) by taking the natural logarithm converts it into a product g(x)·ln(f(x)), which is often easier to analyze. After finding the limit of the logarithm, exponentiate the result to obtain the original limit.

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Percorso guidato

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Intro to Transformations

L'Hôpital's Rule

L'Hôpital's Rule helps evaluate limits that result in indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator separately. This technique is often used after logarithmic transformation to find limits involving powers.

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