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

2:47 minutes

Problem 35

Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→4 x^2 − 16 / 4 − x

Verified step by step guidance

The expression \( \frac{x^2 - 16}{4 - x} \) is in an indeterminate form \( \frac{0}{0} \) as \( x \to 4 \).

Notice that \( x^2 - 16 \) is a difference of squares, which can be factored as \( (x - 4)(x + 4) \).

Substitute the factored form into the expression: \( \frac{(x - 4)(x + 4)}{4 - x} \).

Notice that \( 4 - x = -(x - 4) \), so the expression becomes \( \frac{(x - 4)(x + 4)}{-(x - 4)} \). Cancel \( x - 4 \) from the numerator and denominator, leaving \( -(x + 4) \).

Now that the expression is simplified to \( -(x + 4) \), substitute \( x = 4 \) to find the limit.>

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.

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 specific points, which is crucial for evaluating expressions that may be undefined at those points. In this case, we are interested in the limit as x approaches 4.

Factoring

Factoring is the process of breaking down an expression into simpler components, which can help simplify complex limits. In the given limit, the expression x^2 - 16 can be factored into (x - 4)(x + 4), allowing for cancellation of terms that may lead to an indeterminate form when directly substituting x = 4.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Recognizing these forms is essential, as they often require further manipulation, such as factoring or applying L'Hôpital's Rule, to evaluate the limit correctly.

Watch next

Master Finding Limits by Direct Substitution with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question