Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→−5⁻ (3x) / (2x + 10)

Verified step by step guidance

1. Identify the type of limit: This is a one-sided limit as x approaches -5 from the left, denoted as x → -5⁻.
2. Analyze the denominator: As x approaches -5 from the left, the expression 2x + 10 approaches 0. Specifically, since x is slightly less than -5, 2x + 10 will be slightly negative.
3. Analyze the numerator: The numerator 3x approaches 3(-5) = -15 as x approaches -5.
4. Determine the sign of the fraction: Since the numerator approaches -15 (a negative value) and the denominator approaches a small negative value, the overall fraction (3x)/(2x + 10) approaches a positive value.
5. Conclude the limit: As x approaches -5 from the left, the fraction (3x)/(2x + 10) approaches positive infinity, denoted as ∞.

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. Limits can be finite or infinite, and they are essential for defining derivatives and integrals.

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, either the left (denoted as x→a⁻) or the right (denoted as x→a⁺). This concept is crucial when dealing with functions that may have different behaviors on either side of a point, particularly at points of discontinuity.

Infinite Limits

An infinite limit occurs when the value of a function increases or decreases without bound as the input approaches a certain point. This is indicated by writing the limit as ∞ or −∞. Understanding infinite limits is important for analyzing vertical asymptotes and the overall behavior of functions near points where they may not be defined.