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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.15
Chapter 2, Problem 2.15

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

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1
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.
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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.
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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.
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Related Practice
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Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

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Which of the following statements are correct? Choose all that apply.


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Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

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Determine the following limits.


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