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Chapter 2: Problem 17

Find the limit, if it exists. $$ \lim _{x \rightarrow x} \frac{-x^{3}+2 x}{2 x^{2}-3} $$

Short Answer

Expert verified
The limit is defined as the expression given itself because the setup is inconsistent without extra context.

Step by step solution

01

Understand the Limit Expression

We need to find the limit \( \lim \limits _{x \rightarrow x} \frac{-x^{3}+2x}{2x^{2}-3} \). This problem involves evaluating the expression as \( x \) approaches itself, which simplifies checking for any indeterminate forms.
02

Simplify the Fraction

Notice that there is a misunderstanding here. In this case, the limit seems to imply \( \lim _{x \rightarrow c} \) with an unspecified \( c \), as there's no typical limit form presented such as \( \lim _{x \rightarrow 0} \). The expression as written effectively returns itself when \( x \) is replaced, given no approaching value is stated unambiguously.
03

Conclude on the Setup

Presuming a misunderstanding in notation, and that an unspecified condition was intended, the limit at any specific point could be analyzed if defined separately. Common evaluation could potentially involve known limits or calculus techniques. However, based solely on the given expression, and without additional condition, evaluating directly as written is not necessary because it reduces to substituting the general expression term-wise.

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Key Concepts

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

Limit Evaluation
Limit evaluation is a fundamental concept in calculus that helps us understand the behavior of functions as they approach a particular value. When evaluating limits, we are essentially asking, "What value does the function get close to as the input approaches a certain number?" If the function approaches a specific number, the limit exists. To find this limit, it's important to carefully analyze the given mathematical expression:
  • Identify the form of the limit. Standard forms include approaching a number or infinity.
  • Substitute the value, if possible, to find the limit directly.
  • If substitution leads to an undefined form, reevaluate using algebraic manipulation.
Evaluating limits can sometimes lead to special cases known as indeterminate forms, which require deeper exploration to resolve.
Indeterminate Forms
Indeterminate forms occur when a limit expression yields an undefined or contradictory result upon direct substitution. One classic example is the form \( \frac{0}{0} \). When you see this, it means that both the numerator and denominator approach zero, creating uncertainty about the overall value. Other common indeterminate forms include \( \frac{\infty}{\infty} \), \( 0 \times \infty \), \( \infty - \infty \), among others.
To resolve indeterminate forms, you can use several techniques:
  • Factor and simplify the expression if possible.
  • Apply L'H么pital's rule, which involves differentiating the numerator and denominator.
  • Edit the expression to strategic equivalents that are easier to evaluate.
These tools help convert the problematic forms into determinate ones, allowing for accurate limit computation.
Calculus Techniques
Calculus techniques are tools and strategies used to solve limits and other calculus problems efficiently. These techniques take advantage of algebraic manipulation, geometric interpretation, and derivative applications. Here are a few techniques particularly useful for limit evaluation:
  • Algebraic Manipulation: Simplify complex expressions by factoring and cancelling out common terms.
  • Mentioned Rule: L'H么pital's rule is specifically utilized to deal with indeterminate forms by differentiating both relevant parts of the fraction.
  • Substitution: Temporarily replacing the variable with a new parameter can sometimes clarify complex expressions.
Understanding and applying these techniques can break down seemingly complicated problems into more manageable parts. Mastering these tools ensures a student can confidently tackle any limit evaluation question they encounter.

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Most popular questions from this chapter

Charles' law for gases states that if the pressure remains constant, then the relationship between the volume \(V\) that a gas occupies and its temperature \(T\) (in \(\left.{ }^{\circ} \mathrm{C}\right)\) is given by \(V=V_{0}\left(1+\frac{1}{273} T\right)\). The temperature \(T=-273^{\circ} \mathrm{C}\) is absolute zero. (a) Find \(\lim _{T \rightarrow-273^{*}} V\). (b) Why is a right-hand limit necessary?

Verify the intermediate value theorem (2.26) for \(f\) on the stated interval \([a, b]\) by showing that if \(f(a) \leq w \leq f(b),\) then \(f(c)=w\) for some \(c\) in \([a, b].\) $$ f(x)=x^{3}+1: \quad[-1,2] $$

Show that \(f\) is continuous on the given interval. \(f(x)=\sqrt{16-x} ; \quad(-\infty, 16]\)

Exercise \(27-32:\) Sketch the graph of tie piecewise-defined function \(f\) and, for the indicated value of \(a\), find each limit, if it exists: (a) \(\lim f(x)\) (b) \(\lim f(x)\) |c) \(\lim f(x)\) $$ f(x)=\left\\{\begin{array}{ll} 3 x & \text { if } x \leq 2 \\ x^{2} & \text { if } x>2 \end{array} \quad a=2\right. $$

Use the sandwich theorem to verify the limit. $$ \begin{aligned} &\lim _{x \rightarrow 0} \frac{|x|}{\sqrt{x^{4}+4 x^{2}+7}}=0\\\ &\text { (Hint: Use } f(x)=0 \text { and } g(x)=|x| \text { .) } \end{aligned} $$
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