Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)
4. Applications of Derivatives
Differentials
3:17 minutesProblem 4.7.43
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ -1 (x³ - x² - 5x - 3)/(x⁴ + 2x³ - x² -4x -2)
Verified step by step guidance1
First, substitute x = -1 into the expression to check if the limit results in an indeterminate form like 0/0 or ∞/∞. Calculate the numerator and denominator separately at x = -1.
If the substitution results in an indeterminate form, l'Hôpital's Rule can be applied. This rule states that for limits of the form 0/0 or ∞/∞, the limit of the ratio of the derivatives of the numerator and denominator can be used.
Differentiate the numerator, f(x) = x³ - x² - 5x - 3, with respect to x. The derivative is f'(x) = 3x² - 2x - 5.
Differentiate the denominator, g(x) = x⁴ + 2x³ - x² - 4x - 2, with respect to x. The derivative is g'(x) = 4x³ + 6x² - 2x - 4.
Evaluate the limit of the new expression lim_x→ -1 f'(x)/g'(x) using the derivatives found in the previous steps. Substitute x = -1 into the derivatives to find the limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches -1 involves determining the behavior of the function near that point, which may require simplification or algebraic manipulation.
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L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits when direct substitution leads to indeterminate results.
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Polynomial Functions
Polynomial functions are expressions involving variables raised to whole number powers, combined using addition, subtraction, and multiplication. In the given limit problem, both the numerator and denominator are polynomials. Understanding their behavior, such as their degree and leading coefficients, is crucial for evaluating limits, especially as they approach specific values or infinity.
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