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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 2.15
Capitolo 2, Problema 2.15

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

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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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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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