Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.


lim (x²/2 − 1/x) as


d. x→−1

Verified step by step guidance

1. Identify the function for which you need to find the limit: \( f(x) = \frac{x^2}{2} - \frac{1}{x} \).
2. Substitute \( x = -1 \) into the function to evaluate the limit directly: \( f(-1) = \frac{(-1)^2}{2} - \frac{1}{-1} \).
3. Calculate each term separately: \( \frac{(-1)^2}{2} = \frac{1}{2} \) and \( -\frac{1}{-1} = 1 \).
4. Combine the results from the previous step: \( \frac{1}{2} + 1 \).
5. Simplify the expression to find the limit as \( x \to -1 \).

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near specific points, including points of discontinuity or infinity. Evaluating limits is essential for determining the continuity of functions and for finding derivatives and integrals.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. Understanding rational functions is crucial for limit evaluation, especially when determining behavior as the variable approaches a specific value. In this case, the function involves a polynomial in the numerator and a term involving x in the denominator, which can affect the limit as x approaches -1.

Substitution in Limits

Substitution is a technique used in limit evaluation where you directly replace the variable with the value it approaches. This method is effective when the function is continuous at that point. However, if direct substitution leads to an indeterminate form, further analysis or algebraic manipulation may be necessary to find the limit.