Find a value of the constant k such that the limit exists. limx →-∞ (e^2 x-5)/(e^k x+3)

Name: Problem 50, Chapter 1: Calculus
Uploaded: 2021-08-03T20:13:45-08:00
Duration: 1 min 13 s
Channel: Carson Merrill
Description: Find a value of the constant k such that the limit exists. limx →-∞ (e^2 x-5)/(e^k x+3)

step 1 But the given problem, we want to find the value of the constant k, such that the limit exists.

Find a value of the constant k such that the limit exists....

Key Concepts

Exponential Function Behavior

Exponential functions exhibit rapid change either in growth or decay depending on the sign of the exponent. When the exponent is multiplied by a positive constant, as x approaches negative infinity, the function decays at a rate that depends on that constant. This understanding is pivotal in analyzing limits where exponential terms are present.

Limits at Infinity

Limiting behavior as x approaches infinity or negative infinity requires identifying which parts of an expression dominate the behavior of the function. For functions that involve exponentials, the dominating term is determined by the exponent’s coefficient, which influences how fast the term decays or grows.

Dominant Term Analysis

When a function is composed of several terms, especially in a ratio, the term with the slowest decay or fastest growth controls the overall behavior of the limit. By comparing the rates of decay in both the numerator and the denominator, one can determine conditions on parameters that ensure the limit exists.

Parameter Conditions in Asymptotic Analysis

Often, the existence of a limit depends on specific values of parameters within the function. Analyzing the asymptotic behavior can reveal necessary constraints on these parameters, ensuring that the contributions from each term balance appropriately for the limit to be finite and well-defined.

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