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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2:06 minutes

Problem 2.15

Textbook Question

Determine the following limits.
lim x→1 x^3 − 7x^2 + 12x / 4 − x

Verified step by step guidance

Identify the limit expression: \( \lim_{{x \to 1}} \frac{x^3 - 7x^2 + 12x}{4 - x} \).

Check if direct substitution of \( x = 1 \) results in an indeterminate form. Substitute \( x = 1 \) into the numerator and denominator.

Since direct substitution results in an indeterminate form \( \frac{0}{0} \), apply algebraic manipulation to simplify the expression. Factor the numerator \( x^3 - 7x^2 + 12x \).

Factor out \( x \) from the numerator: \( x(x^2 - 7x + 12) \). Further factor \( x^2 - 7x + 12 \) into \( (x - 3)(x - 4) \).

Rewrite the expression as \( \lim_{{x \to 1}} \frac{x(x - 3)(x - 4)}{4 - x} \) and simplify by canceling common factors, then evaluate the limit.

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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 specific points, which is crucial for evaluating functions that may not be defined at those points. In this case, we are interested in the limit as x approaches 1.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This technique is often used to simplify expressions, especially when evaluating limits, as it can help eliminate indeterminate forms like 0/0. In the given limit, factoring the numerator will be essential to simplify the expression before substituting x = 1.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to an undefined expression, such as 0/0 or ∞/∞. Recognizing these forms is crucial because they indicate that further analysis, such as factoring or applying L'Hôpital's Rule, is needed to evaluate the limit correctly. In this problem, substituting x = 1 initially results in an indeterminate form.