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


lim_x→ 0 (eˣ - 1) / (x² + 3x)

Verified step by step guidance

1. First, substitute x = 0 into the limit expression to check if it results in an indeterminate form. Substituting gives (e^0 - 1) / (0^2 + 3*0) = 0/0, which is an indeterminate form.
2. Since the limit is in the indeterminate form 0/0, we can apply l'Hôpital's Rule. This rule 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, provided the derivatives exist.
3. Differentiate the numerator and the denominator separately. The derivative of the numerator, e^x - 1, is e^x. The derivative of the denominator, x² + 3x, is 2x + 3.
4. Apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→0 (e^x) / (2x + 3).
5. Substitute x = 0 into the new expression to evaluate the limit: (e^0) / (2*0 + 3) = 1/3. Thus, the limit is 1/3.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for evaluating expressions that may be indeterminate, such as 0/0 or ∞/∞. In this question, we need to analyze the behavior of the function as x approaches 0 to determine the limit.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) yields an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This technique simplifies the evaluation of complex limits, making it particularly useful in this problem.

Exponential Functions

Exponential functions, such as eˣ, are functions where a constant base is raised to a variable exponent. They exhibit unique properties, including the fact that eˣ approaches 1 as x approaches 0. Understanding the behavior of exponential functions is essential for evaluating limits involving them, as they often lead to indeterminate forms that require careful analysis.