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.

Verified video answer for a similar problem:

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

Video duration:

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.

Recommended video:

3:56

Slope-Intercept Form

Related Practice

Textbook Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.

$f (x) = \sin x$

Textbook Question

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

lim x→0 x^2=0 (Hint: Use the identity √x2=|x|.)

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→3 x − 3 /|x − 3|

views

Textbook Question

Evaluate each limit and justify your answer.

lim x→2 (3 / 2x^5−4x^2−50)^4

views