Question

Find the limits. $$ \lim _{x \rightarrow 1} \frac{3 x^{2}-x-2}{2 x^{2}+x-3} $$

   Find the limits.
$$
\lim _{x \rightarrow 1} \frac{3 x^{2}-x-2}{2 x^{2}+x-3}
$$

Calculus Early Transcendentals

Calculus Early Transcendentals

Howard Anton, Irl… 9th Edition

Chapter 1, Problem 12

Instant Answer

Solved by Expert R M

08/14/2020

Step 1

The numerator $3x^2 - x - 2$ can be factored into $(3x+2)(x-1)$ and the denominator $2x^2 + x - 3$ can be factored into $(2x-1)(x+3)$. So, the expression becomes: $$ \lim _{x \rightarrow 1} \frac{(3x+2)(x-1)}{(2x-1)(x+3)} $$ Show more…

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Find the limits. $$ \lim _{x \rightarrow 1} \frac{3 x^{2}-x-2}{2 x^{2}+x-3} $$

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Key Concepts

Limits of Functions

Limits help us understand the behavior of a function as the input approaches a specific value, even if the function is not defined at that point. This concept is fundamental in calculus for determining continuity, derivatives, and integrals, and it often involves looking for ways to resolve situations where direct substitution does not immediately yield a clear result.

Indeterminate Forms

Indeterminate forms, such as 0/0, occur when direct substitution in a limit yields ambiguous or undefined results. Recognizing these forms is essential because they signal that standard evaluation methods may fail, and alternative techniques, such as algebraic manipulation, are required to compute the limit.

Algebraic Simplification

Algebraic simplification involves techniques, including factoring and cancellation, to rewrite complex expressions into simpler forms. This step is particularly useful in limit calculations where indeterminate forms appear, enabling the cancellation of common factors and the resolution of the limit through more straightforward expressions.

Factoring Techniques

Factoring techniques refer to methods used to express polynomials as products of their factors. By factoring both the numerator and the denominator in a rational expression, one can often identify common factors that cause indeterminate forms, allowing for their cancellation so that the limit can be evaluated by direct substitution.

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