Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→∞ √(x² + 1) / (x + 1)

검증된 단계별 안내

Identify the highest power of x in the denominator, which is x in this case.

Divide both the numerator and the denominator by x, the highest power of x in the denominator.

Rewrite the expression: \( \frac{\sqrt{x^2 + 1}}{x + 1} = \frac{\sqrt{x^2 + 1}/x}{(x + 1)/x} \).

Simplify the expression: \( \frac{\sqrt{1 + \frac{1}{x^2}}}{1 + \frac{1}{x}} \).

Evaluate the limit as x approaches infinity: As x → ∞, \( \frac{1}{x^2} \to 0 \) and \( \frac{1}{x} \to 0 \), so the expression simplifies to \( \frac{\sqrt{1 + 0}}{1 + 0} = 1 \).

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Limits at Infinity

Limits at infinity involve finding the behavior of a function as the input approaches positive or negative infinity. This concept helps determine the end behavior of functions, particularly rational functions, by analyzing how the terms grow or shrink as x becomes very large or very small. Understanding limits at infinity is crucial for evaluating the asymptotic behavior of functions.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. To find limits involving rational functions, especially as x approaches infinity, it is often useful to divide both the numerator and the denominator by the highest power of x present in the denominator. This simplification helps identify dominant terms and facilitates the calculation of limits.

Simplification Techniques

Simplification techniques involve algebraic manipulation to make complex expressions easier to evaluate. In the context of limits, dividing by the highest power of x in the denominator is a common technique to simplify the expression, allowing us to focus on the dominant terms that dictate the limit behavior. This approach is essential for handling noninteger or negative powers of x effectively.