Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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2:30 minutes

Problem 28

Textbook Question

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

Verified step by step guidance

Identify the type of limit: This is a one-sided limit as \( x \) approaches \(-5\) from the right (denoted as \( x \to -5^+ \)).

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.

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.

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.

Conclude that the limit is \(-\infty\) as \( x \to -5^+ \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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^+.

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.