Determine the following limits.
lim x→−5^+ x − 5 / x + 5

Verified step by step guidance

1. Identify the type of limit: This is a one-sided limit as \( x \) approaches \(-5\) from the right (denoted as \( x \to -5^+ \)).
2. Substitute \( x = -5 \) into the expression \( \frac{x - 5}{x + 5} \) to check for any indeterminate forms. This gives \( \frac{-5 - 5}{-5 + 5} = \frac{-10}{0} \), indicating a division by zero.
3. Analyze the behavior of the numerator and denominator as \( x \to -5^+ \): The numerator \( x - 5 \) approaches \(-10\), and the denominator \( x + 5 \) approaches \(0\) from the positive side.
4. Determine the sign of the expression as \( x \to -5^+ \): Since the numerator is negative and the denominator approaches zero from the positive side, the overall expression approaches negative infinity.
5. Conclude that the limit is \(-\infty\) as \( x \to -5^+ \).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit as x approaches -5 from the right, denoted as x → -5^+.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only. The notation x → -5^+ indicates that we are considering values of x that are greater than -5. This is crucial for determining the limit in cases where the function may behave differently from the left side compared to the right side.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In this limit problem, the expression (x - 5) / (x + 5) is a rational function. Understanding how to simplify and evaluate limits involving rational functions is essential, especially when determining behavior near points where the denominator may approach zero.