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

lim_x→ 1 (x² + 2x) / (x +3)

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First, substitute x = 1 into the expression (x² + 2x) / (x + 3) to check if the limit results in an indeterminate form like 0/0.

After substitution, if the expression results in 0/0, l'Hôpital's Rule can be applied. This rule states that for limits of the form 0/0 or ∞/∞, the limit of f(x)/g(x) as x approaches a value can be found by taking the limit of f'(x)/g'(x).

Differentiate the numerator f(x) = x² + 2x to get f'(x) = 2x + 2.

Differentiate the denominator g(x) = x + 3 to get g'(x) = 1.

Apply l'Hôpital's Rule by taking the limit of the new expression (2x + 2) / 1 as x approaches 1, and evaluate this 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 value at points where it may not be explicitly defined. Evaluating limits is crucial 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 ∞/∞. It 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.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Continuous functions do not have breaks, jumps, or holes, making them easier to analyze when evaluating limits. Understanding continuity is essential for applying l'Hôpital's Rule effectively.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→ 1 (x² + 2x) / (x +3)