Use l’Hôpital’s rule to find the limits in Exercises 7–52.

9. lim (t → -3) (t³ - 4t + 15) / (t² - t - 12)

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First, identify the limit expression: \(\lim_{t \to -3} \frac{t^{3} - 4t + 15}{t^{2} - t - 12}\).

Evaluate the numerator and denominator separately at \(t = -3\) to check if the limit is an indeterminate form. Calculate \((-3)^{3} - 4(-3) + 15\) and \((-3)^{2} - (-3) - 12\).

If both numerator and denominator evaluate to 0, then the limit is of the form \(\frac{0}{0}\), and l’Hôpital’s Rule can be applied.

Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(t\): find \(\frac{d}{dt}(t^{3} - 4t + 15)\) and \(\frac{d}{dt}(t^{2} - t - 12)\).

After differentiation, substitute \(t = -3\) into the new expression \(\frac{\frac{d}{dt}(\text{numerator})}{\frac{d}{dt}(\text{denominator})}\) to find the 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 particular value. When direct substitution results in an indeterminate form like 0/0 or ∞/∞, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.

l’Hôpital’s Rule

l’Hôpital’s rule provides a method to evaluate limits that yield indeterminate forms by differentiating the numerator and denominator separately and then taking the limit of their quotient. It applies only when the original limit results in 0/0 or ∞/∞.

Differentiation of Polynomial Functions

Differentiation involves finding the derivative of a function, which represents its rate of change. For polynomials, derivatives are found by applying the power rule term-by-term, which is essential when using l’Hôpital’s rule to differentiate numerator and denominator.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.9. lim (t → -3) (t³ - 4t + 15) / (t² - t - 12)