Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)

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First, substitute x = -1 into the expression to check if the limit results in an indeterminate form. Calculate the numerator: (-1)^4 + (-1)^3 + 2(-1) + 2, and the denominator: (-1) + 1.

Notice that both the numerator and the denominator evaluate to 0, indicating an indeterminate form 0/0. This suggests that l'Hôpital's Rule can be applied.

Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit is the same as the limit of f'(x)/g'(x) as x approaches that value. Differentiate the numerator: d/dx(x^4 + x^3 + 2x + 2) and the denominator: d/dx(x + 1).

The derivative of the numerator is 4x^3 + 3x^2 + 2, and the derivative of the denominator is 1. Substitute these derivatives back into the limit expression: lim_x→ -1 (4x^3 + 3x^2 + 2) / 1.

Evaluate the limit by substituting x = -1 into the new expression: 4(-1)^3 + 3(-1)^2 + 2. Calculate the result to find the limit.

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is essential for determining continuity, derivatives, and integrals.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit. This technique simplifies the process of finding limits in complex cases.

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the context of limits, understanding the behavior of polynomial functions as they approach specific values is crucial, especially since they are continuous and differentiable everywhere. This knowledge aids in simplifying expressions before applying limit evaluation techniques.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→ -1 (x⁴ + x³ + 2x + 2) / (x + 1)