Problem 21 Find the limits in Exercises \(1... [FREE SOLUTION]

Chapter 2: Problem 21

Find the limits in Exercises $17-22$. $$ \begin{array}{l}{\lim \frac{x^{2}-3 x+2}{x^{3}-2 x^{2}} \text { as }} \\\ {\begin{array}{ll}{\text { a. } x \rightarrow 0^{+}} & {\text { b. } x \rightarrow 2^{+}} \\ {\text { c. } x \rightarrow 2^{-}} & {\text { d. } x \rightarrow 2} \\ {\text { e. What, if anything, can be said about the limit as } x \rightarrow 0 ?}\end{array}}\end{array} $$

Short Answer

Expert verified

a) \$ -\infty \$ b) DNE c) DNE d) DNE e) \$ -\infty \$

Step by step solution

Simplify the Expression

The given expression is $ \frac{x^{2} - 3x + 2}{x^{3} - 2x^{2}} $. To simplify, notice that both the numerator and the denominator can be factored. The numerator factors to $(x-1)(x-2)$ and the denominator factors to $x^{2}(x-2)$. The expression becomes $ \frac{(x-1)(x-2)}{x^2(x-2)} $. Cancel the common factor $(x-2)$ to get $ \frac{x-1}{x^2} $ for $x eq 2$.

Limit as $x \rightarrow 0^{+}$

Substitute $x = 0$ into the simplified expression $ \frac{x-1}{x^2} $. This becomes $ \frac{0-1}{0^2} = \frac{-1}{0} $, which suggests the limit goes towards negative infinity. Hence, $ \lim_{x \rightarrow 0^{+}} \frac{x - 1}{x^2} = -\infty $.

Limit as $x \rightarrow 2^{+}$

Using the original expression, check the behavior as $ x \approaches 2^{+} $. The denominator $x^2 (x-2)$ approaches zero from a positive side and the numerator approaches zero. However, since the cancelled term $(x-2)$ was initially in both the numerator and denominator, they both approach 0 from the same direction. Therefore, as $x \approaches 2^{+}$, the limit does not exist as a finite number but rather approaches +\infty.

Limit as $x \rightarrow 2^{-}$

Similarly to part (b), evaluate the behavior as $ x \approaches 2^{-} $. The expression behaves the same way as when $ x \approaches 2^{+} $, hence, the limit is also +\infty since both the numerator and denominator approach zero but with positive sign on tiny margin around point 2.

Limit as $x \rightarrow 2$

As $ x \approaches 2$, the limit reflects an undefined point since the canceled $(x-2)$ leads to division by zero; thus, technically the limit does not exist across the point precisely at $x=2$.

Revisit Limit as $x \rightarrow 0$ from prior Simplification

The limit as $x \rightarrow 0$ need thorough understanding of behavior of bounding expressions. As previously derived in section from right side, $ \lim_{x \rightarrow 0} \frac{x - 1}{x^2} = -\infty $. Therefore, facing similar limits from left and right.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Evaluation

When tackling calculus problems involving limits, the primary goal is to determine the behavior of a function as the variable approaches a specific point. For the exercise given, various approaches of limit evaluation can be explored:

At specific points, like where the function was evaluated for $x \rightarrow 0^{+}$, one can directly substitute and observe the behavior of both the numerator and denominator. If direct substitution leads to an undefined form, like $\frac{-1}{0}$, then typically, infinite limits might occur.
In cases like $x \rightarrow 2$, a different method applies, as direct substitution yields division by zero. Such a scenario leads to analyzing how the numerator and denominator approach zero simultaneously.

Always begin by checking if direct substitution is possible. If limits do not comfortably yield an answer directly, simplification and further analytical methods are necessary.

Function Simplification

Simplifying a function is often the first step before evaluating its limit, mainly if a complex expression is present. Let's break down what we did for the problem:

The expression $\frac{x^2 - 3x + 2}{x^3 - 2x^2}$ contains a rational function, where both the numerator and the denominator can be factored. The numerator factors into $(x-1)(x-2)$ and the denominator into $x^2(x-2)$.
By recognizing common factors in both the numerator and the denominator, namely $(x-2)$, one can cancel them, simplifying the function to $\frac{x-1}{x^2}$ for $x eq 2$.

This simplification makes limit calculations more straightforward, often transforming complicated expressions into more manageable forms. Carefully simplifying functions can reveal fundamental behaviors that are otherwise hidden in their unsimplified forms.

Behavior Analysis at Points

The behavior of functions at specific points is crucial for understanding the nature of limits. Several steps help elucidate this process:

For $x \rightarrow 0$, behavior analysis showed that the function $\frac{x-1}{x^2}$ becomes significantly large negatively, indicating a tendency towards $-\infty$.
As $x \rightarrow 2$, analyzing the function before and after cancellation of common terms provides insights into the limit behavior. Specifically, as $x$ approaches 2 from either side, the cancelled term causes both parts of the fraction to head towards zero, reinforcing the need for careful behavior analysis.
After simplification, if division by zero occurs at a specified point, as seen here for $x=2$, it typically indicates an undefined limit rather than a simple numerical value.

Understanding these behaviors is essential, especially when dealing with apparent discontinuities or undefined points in a function's domain. This analysis forms the basis for predicting and articulating how functions behave near critical points.

91影视

Short Answer

Step by step solution

Simplify the Expression

Limit as \(x \rightarrow 0^{+}\)

Limit as \(x \rightarrow 2^{+}\)

Limit as \(x \rightarrow 2^{-}\)

Limit as \(x \rightarrow 2\)

Revisit Limit as \(x \rightarrow 0\) from prior Simplification

Key Concepts

Limit Evaluation

Function Simplification

Behavior Analysis at Points

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Statistics

Mechanics Maths

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.