Question

Evaluate: $\lim _{x \rightarrow 1} \frac{x^2+x-2}{x^2-1}$

    Evaluate: $\lim _{x \rightarrow 1} \frac{x^2+x-2}{x^2-1}$

Calculus With Trigonometry and Analytic Geometry

Calculus With Trigonometry and Analytic Geometry

John Saxon, Frank… 1st Edition

Chapter 32, Problem 16

Instant Answer

Solved by Expert Aditya Sood

03/02/2020

Step 1

Substitute \( x = 1 \) into the expression \(\frac{x^2+x-2}{x^2-1}\). \[ \frac{1^2 + 1 - 2}{1^2 - 1} = \frac{0}{0} \] This is an indeterminate form, so we need to simplify the expression. Show more…

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Evaluate: $\lim _{x \rightarrow 1} \frac{x^2+x-2}{x^2-1}$

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Key Concepts

Removable Discontinuities

Removable discontinuities occur in rational functions when a factor causing a zero in the denominator also appears in the numerator. By canceling this common factor, the function's expression can be simplified, 'removing' the discontinuity and allowing the limit to be calculated using direct substitution.

Factoring Polynomials

Factoring polynomials is a key algebraic technique used to simplify expressions, including those found in limits. By rewriting polynomials as products of simpler expressions, common factors in the numerator and denominator can be canceled, resolving indeterminate forms and making the limit easier to evaluate.

Indeterminate Forms

Indeterminate forms occur when substituting a value into a function results in expressions like 0/0. These forms require additional algebraic manipulation, such as factoring, to simplify the expression and evaluate the limit properly.

Limits

Limits are a fundamental concept in calculus used to describe the behavior of a function as its input approaches a specific value. They provide a way to understand the behavior of functions when direct substitution might not be possible due to a discontinuity or an indeterminate form.

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