Find each limit algebraically. limx →1 (x^3-x^2+x-1)/(x^4-x^3+2 x-2)

Name: Problem 37, Chapter 13: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2021-06-15T18:56:55-08:00
Duration: 1 min 36 s
Channel: Dwijendra Rao
Description: Find each limit algebraically. limx →1 (x^3-x^2+x-1)/(x^4-x^3+2 x-2)

step 1 Hello, we have to find the limit x tends to 1 x cube minus x square plus x minus 1 divided by x

Question

Find each limit algebraically. $$ \lim _{x \rightarrow 1} \frac{x^3-x^2+x-1}{x^4-x^3+2 x-2} $$

   Find each limit algebraically.
$$
\lim _{x \rightarrow 1} \frac{x^3-x^2+x-1}{x^4-x^3+2 x-2}
$$

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan 4th Edition

Chapter 13, Problem 37

Instant Answer

Solved by Expert Dwijendra Rao

06/15/2021

Step 1

First, we can factor the numerator and denominator: $$ \frac{x^3-x^2+x-1}{x^4-x^3+2x-2} = \frac{x^2(x-1) + (x-1)}{x^3(x-1) + 2(x-1)} $$ Now we can factor out the common factor of (x-1) from both the numerator and denominator: $$ \frac{(x-1)(x^2+1)}{(x-1)(x^3+2)} Show more…

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Find each limit algebraically. $$ \lim _{x \rightarrow 1} \frac{x^3-x^2+x-1}{x^4-x^3+2 x-2} $$

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Key Concepts

Indeterminate Forms

Indeterminate forms occur when directly substituting the limit value into an expression yields an ambiguous expression like 0/0. Recognizing a 0/0 form indicates that further algebraic manipulation, such as factoring or canceling common terms, is necessary to correctly evaluate the limit.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as the product of simpler polynomials or factors. This is particularly useful in limit problems to identify and cancel common factors in the numerator and denominator, which can eliminate discontinuities and simplify the expression.

Limit Evaluation Techniques

Limit evaluation techniques are strategies used to determine the value a function approaches as the independent variable nears a specific point. Techniques such as factoring, canceling common factors, and applying the properties of limits help resolve cases where a direct substitution leads to an indeterminate form.

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