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

Previous problemNext problem

2:31 minutes

Problem 11

Textbook Question

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 f(x) / g(x)−h(x)

Verified step by step guidance

Identify the given limits: \( \lim_{{x \to 1}} f(x) = 8 \), \( \lim_{{x \to 1}} g(x) = 3 \), and \( \lim_{{x \to 1}} h(x) = 2 \).

Recognize that the problem requires finding \( \lim_{{x \to 1}} \frac{f(x)}{g(x) - h(x)} \).

Apply the limit law for quotients: \( \lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)} \) provided \( \lim_{{x \to a}} g(x) \neq 0 \).

Calculate the limit of the denominator: \( \lim_{{x \to 1}} (g(x) - h(x)) = \lim_{{x \to 1}} g(x) - \lim_{{x \to 1}} h(x) = 3 - 2 = 1 \).

Use the quotient limit law: \( \lim_{{x \to 1}} \frac{f(x)}{g(x) - h(x)} = \frac{\lim_{{x \to 1}} f(x)}{\lim_{{x \to 1}} (g(x) - h(x))} = \frac{8}{1} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In this case, we have limits for functions f(x), g(x), and h(x) as x approaches 1. Understanding limits is crucial for evaluating expressions involving functions at specific points.

Limit Laws

Limit laws are rules that allow us to compute the limits of functions based on the limits of their components. For example, the quotient law states that if the limits of f(x) and g(x) exist and g(x) is not zero at the limit point, then lim x→c (f(x)/g(x)) = lim x→c f(x) / lim x→c g(x). These laws are essential for simplifying and calculating complex limits.

Algebraic Manipulation of Limits

Algebraic manipulation involves rearranging and simplifying expressions to facilitate limit evaluation. In the given problem, we need to compute the limit of the expression f(x)/g(x) - h(x). This requires applying limit laws and performing algebraic operations to find the limit of the entire expression as x approaches 1.