In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

5. lim (x → 0) (1 - cos x) / x²

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

Check if the limit is an indeterminate form by substituting \(x = 0\): numerator becomes \(1 - \cos 0 = 1 - 1 = 0\), denominator becomes \(0^{2} = 0\), so the form is \(\frac{0}{0}\), which allows the use of l'Hôpital's Rule.

Apply l'Hôpital's Rule by differentiating the numerator and denominator separately: differentiate numerator \(\frac{d}{dx}(1 - \cos x) = \sin x\), differentiate denominator \(\frac{d}{dx}(x^{2}) = 2x\).

Rewrite the limit using the derivatives: \(\lim_{x \to 0} \frac{\sin x}{2x}\).

Evaluate this new limit by substituting \(x = 0\) or by using known limits from Chapter 2, such as \(\lim_{x \to 0} \frac{\sin x}{x} = 1\), to find the value of the limit.

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Key Concepts

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

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 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. This rule simplifies complex limits by differentiating numerator and denominator separately.

Limit of a Function as x Approaches a Point

The limit of a function as x approaches a point describes the value that the function approaches near that point. Understanding how to evaluate limits, especially when direct substitution leads to indeterminate forms, is fundamental in calculus. It helps analyze the behavior of functions near specific values.

Using Series Expansion to Evaluate Limits

Series expansion, such as the Taylor or Maclaurin series, expresses functions as infinite sums of terms based on derivatives at a point. For small values of x, approximating functions like cosine with their series helps simplify limits without using l’Hôpital’s Rule. For example, cos x ≈ 1 - x²/2 + x⁴/24 + … near zero.

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