Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
97. lim(x→0) (10^x - 1)/x

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Identify the limit expression: $\lim_{x \to 0} \frac{10^x - 1}{x}$.

Check the form of the limit by substituting $x = 0$: numerator becomes $10^0 - 1 = 1 - 1 = 0$ and denominator is $0$, so the limit is of the form $\frac{0}{0}$, which is indeterminate and allows the use of l'Hôpital's Rule.

Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to $x$: differentiate \$10^x$ using the chain rule, recalling that $\frac{d}{dx} a^x = a^x \ln(a)$ for a positive constant $a$.

Write the new limit after differentiation: $\lim_{x \to 0} \frac{\frac{d}{dx}(10^x - 1)}{\frac{d}{dx} x} = \lim_{x \to 0} \frac{10^x \ln(10)}{1}$.

Evaluate the new limit by substituting $x = 0$: $10^0 \ln(10) = 1 \cdot \ln(10)$, which gives the value of the original limit.

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주요 개념

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Limits and Indeterminate Forms

Limits describe the behavior of a function as the input approaches a certain value. When direct substitution results in an indeterminate form like 0/0, special techniques such as l’Hôpital’s Rule are needed to evaluate the limit.

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that yield indeterminate forms 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.

Derivative of Exponential Functions

The derivative of an exponential function a^x, where a > 0, is given by a^x times the natural logarithm of a (ln a). This rule is essential when applying l’Hôpital’s Rule to limits involving exponential expressions.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.97. lim(x→0) (10^x - 1)/x