Use l’Hôpital’s rule to find the limits in Exercises 7–52.
30. lim (θ → 0) ((1/2)^θ - 1) / θ

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Identify the limit expression: \(\lim_{\theta \to 0} \frac{(\frac{1}{2})^{\theta} - 1}{\theta}\).

Check the form of the limit by substituting \(\theta = 0\): the numerator becomes \((\frac{1}{2})^0 - 1 = 1 - 1 = 0\) and the denominator is \(0\), so the limit is of the indeterminate form \(\frac{0}{0}\).

Since the limit is of the form \(\frac{0}{0}\), apply l'Hôpital's Rule, which states that \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\) if the original limit is indeterminate.

Differentiate the numerator and denominator separately with respect to \(\theta\): the derivative of the numerator \(f(\theta) = (\frac{1}{2})^{\theta} - 1\) is \(f'(\theta) = (\frac{1}{2})^{\theta} \ln(\frac{1}{2})\), and the derivative of the denominator \(g(\theta) = \theta\) is \(g'(\theta) = 1\).

Rewrite the limit using these derivatives: \(\lim_{\theta \to 0} \frac{(\frac{1}{2})^{\theta} \ln(\frac{1}{2})}{1}\), then evaluate this limit by substituting \(\theta = 0\).

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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 forms like 0/0 or ∞/∞, these are called indeterminate forms, requiring special techniques such as l’Hôpital’s rule 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 ∞/∞ by differentiating the numerator and denominator separately and then taking the limit of their quotient. It simplifies complex limits involving functions that are difficult to evaluate directly.

Exponential and Logarithmic Functions

Understanding the behavior and derivatives of exponential functions, such as (1/2)^θ, is essential. These functions can be rewritten using logarithms to facilitate differentiation, which is crucial when applying l’Hôpital’s rule to limits involving exponential expressions.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.30. lim (θ → 0) ((1/2)^θ - 1) / θ