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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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