5-28 Find the limit. limx →∞ (e^3 x-e^-3 x)/(e^3 x+e^-3 x) |...

Name: Problem 26, Chapter 2: Biocalculus Calculus for the Life Sciences
Uploaded: 2019-07-31T22:02:37-08:00
Duration: 3 min 53 s
Channel: Keith Cannon
Description: 5-28 Find the limit. limx →∞ (e^3 x-e^-3 x)/(e^3 x+e^-3 x)

Key Concepts

Limit Evaluation Techniques

Calculating limits of rational functions often involves isolating the dominant behavior of the numerator and denominator. Recognizing which parts of an expression tend to zero or infinity and then simplifying accordingly is a key strategy in calculus for obtaining the limit as a variable approaches infinity.

Exponential Growth and Decay

Exponential functions with a positive exponent, such as e^(3x), grow rapidly as x increases, while those with a negative exponent, like e^(-3x), decay quickly. This contrast is fundamental when evaluating limits, as the large-x behavior of the expression is often dictated by the term that grows or decays the fastest.

Dominant Term in Asymptotic Analysis

In limits involving sums of exponential terms, the term with the highest growth rate becomes overwhelmingly significant as x approaches infinity. Here, the e^(3x) terms in both the numerator and denominator govern the behavior of the entire expression, allowing simplification by neglecting the exponentially decaying terms.

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