Limits of quotients


Find the limits in Exercises 23–42.


limx→−5 (x² + 3x − 10) / x + 5

Verified step by step guidance

1. First, identify the type of limit problem. This is a limit of a quotient as x approaches -5.
2. Check if direct substitution of x = -5 into the expression (x² + 3x − 10) / (x + 5) results in an indeterminate form like 0/0.
3. Substitute x = -5 into the numerator: x² + 3x − 10 becomes (-5)² + 3(-5) − 10.
4. Substitute x = -5 into the denominator: x + 5 becomes -5 + 5.
5. If the substitution results in 0/0, factor the numerator x² + 3x − 10 to see if it can be simplified with the denominator x + 5. Look for common factors to cancel out and resolve the indeterminate form.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms. In this case, evaluating the limit as x approaches -5 requires analyzing the function's behavior around that point.

Quotient of Functions

The quotient of functions involves dividing one function by another. When finding limits of quotients, it is essential to consider the behavior of both the numerator and denominator as the variable approaches a specific value. If the denominator approaches zero while the numerator does not, the limit may be undefined or infinite, necessitating further analysis or simplification.

Factoring and Simplifying

Factoring and simplifying expressions is a crucial technique in calculus, especially when dealing with limits. By factoring the numerator and denominator, one can often cancel common terms, which can resolve indeterminate forms like 0/0. In this problem, factoring the numerator will help in simplifying the expression before evaluating the limit as x approaches -5.