17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)

Verified step by step guidance

1. Identify the form of the limit as x approaches infinity. The expression is (e^(3x)) / (3e^(3x) + 5). As x approaches infinity, both the numerator and the denominator approach infinity, which is an indeterminate form of type ∞/∞.
2. Since the limit is in the indeterminate form ∞/∞, we can apply l'Hôpital's Rule. This rule states that if the limit of f(x)/g(x) as x approaches a value is in the form 0/0 or ∞/∞, then it can be evaluated as the limit of f'(x)/g'(x), provided this new limit exists.
3. Differentiate the numerator and the denominator separately. The derivative of the numerator, e^(3x), with respect to x is 3e^(3x). The derivative of the denominator, 3e^(3x) + 5, with respect to x is 9e^(3x).
4. Apply l'Hôpital's Rule by taking the limit of the new fraction formed by the derivatives: lim_(x→∞) (3e^(3x)) / (9e^(3x)).
5. Simplify the expression by canceling out e^(3x) from the numerator and the denominator, resulting in lim_(x→∞) 3/9. Simplify this fraction to find the limit.

Key Concepts

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 infinity helps determine the behavior of the function at extreme values.

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 in complex expressions.

Exponential Functions

Exponential functions, such as e^x, are functions where a constant base is raised to a variable exponent. They grow rapidly as x increases, which is crucial when evaluating limits at infinity. Understanding their growth behavior helps in determining the dominant terms in expressions, especially when comparing them to polynomial or constant terms.