Find the limit. limx →- ∞ (x+2)/(√(9 x^2+1))

Name: Problem 26, Chapter 2: Calculus
Uploaded: 2021-08-24T23:35:16-08:00
Duration: 1 min 36 s
Channel: Carson Merrill
Description: Find the limit. limx →- ∞ (x+2)/(√(9 x^2+1))

Key Concepts

Limits at Infinity

This concept involves evaluating the behavior of a function as the independent variable grows without bound in either the positive or negative direction. The analysis often focuses on understanding which parts of the function dominate and how they shape the function's end behavior. It is essential in determining asymptotic behavior and long-run tendencies of functions.

Dominant Terms in Polynomial Expressions

When dealing with limits at infinity, particularly in rational functions, the term with the highest degree in the numerator and denominator usually governs the behavior of the entire expression. By focusing on these dominant terms, one can simplify the limit calculations, as lower-order terms become insignificant relative to the dominant ones.

Simplification of Square Root Expressions in Polynomials

This concept entails the process of simplifying expressions that involve the square root of a polynomial by factoring out highest powers. When a square root contains a dominating squared term, it can often be expressed as an absolute value times a constant, which aids in simplifying the limit evaluation process.

Absolute Value Considerations in Limits

In the context of limits, especially when square root expressions simplify to involve absolute values, it becomes important to consider the actual sign of the variable. This is critical when the variable approaches negative infinity, as the absolute value of the variable is then taken as its negation, ensuring accurate evaluation of the limit.

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