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Chapter 1: Problem 3

Find the limit. $$\lim _{x \rightarrow 0} \frac{3 x^{2}}{x^{4}}$$

Short Answer

Expert verified
The limit does not exist; it approaches infinity.

Step by step solution

01

Simplify the Expression

Start by simplifying the fraction \( \frac{3x^2}{x^4} \). This can be done by dividing the numerator and the denominator by \( x^2 \). This gives us:\[\frac{3x^2}{x^4} = \frac{3}{x^2}\]
02

Evaluate the Limit of Simplified Expression

Now that the expression is simplified to \( \frac{3}{x^2} \), evaluate the limit as \( x \) approaches 0:\[\lim_{x \to 0} \frac{3}{x^2}\]As \( x \) approaches 0, \( x^2 \) also approaches 0, making \( \frac{3}{x^2} \) go to infinity since the denominator becomes very small. Thus, \[\lim_{x \to 0} \frac{3}{x^2} = \infty\]
03

Determine if Limit Exists

A limit can be defined only if the expression approaches a particular finite value. Since the limit as \( x \to 0 \) for \( \frac{3}{x^2} \) becomes \( \infty \), the limit does not exist in the context of finite numbers.

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Key Concepts

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

Limit Evaluation
Limit evaluation in calculus is the process of finding the value that a function approaches as the input approaches a certain point. This is a critical tool in calculus for understanding the behavior of functions. For the example given, \[\lim _{x \rightarrow 0} \frac{3 x^{2}}{x^{4}}\], we begin by simplifying the expression.
  • Simplification makes it easier to evaluate limits.
  • Factor out common terms if possible, in this case, \(x^2\) from the numerator and denominator.
  • Simplify to get \(\frac{3}{x^2}\).
Once the expression is simplified, we substitute the approaching point into the simplified expression, in this instance, as \(x\) approaches 0. While direct substitution can sometimes work, other times it results in undefined forms or indeterminate forms, which leads us to our next topic.
Indeterminate Forms
An indeterminate form occurs in calculus when substituting the limiting value directly into a function yields a situation where the limit cannot be easily determined. Common forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), and \(\infty - \infty\).In this problem, the indeterminate form arises because as \(x\) approaches 0, \(\frac{3}{x^2}\) becomes undefined due to division by zero. Here鈥檚 what you should know:
  • It is essential to recognize indeterminate forms to apply strategies to resolve them.
  • Simplifying expressions can often turn an indeterminate form into one that is resolvable.
  • The arithmetic simplification in this exercise results in the expression becoming \(\frac{3}{x^2}\), which approaches infinity, leading to the conclusion that the limit does not exist (in the finite sense).
Understanding indeterminate forms equips us to apply further calculus tools, such as L'H么pital's Rule, where simplification doesn't suffice.
Asymptotic Behavior
Asymptotic behavior describes how a function behaves as the input approaches a particular value, often tending toward infinity or a point of discontinuity. In the expression \(\frac{3}{x^2}\), as \(x\) approaches 0, the function tends toward infinity.
  • Asymptotes are lines that a graph of a function approaches but never touches.
  • Vertical asymptotes occur in fractions when the denominator approaches zero, often leading to infinity.
  • In our expression, as \(x\) gets closer to zero, \(x^2\) gets smaller and smaller, making \(\frac{3}{x^2}\) grow indefinitely.
Recognizing asymptotic behavior helps determine the limit's existence or identify it as not finite or infinite. This understanding is pivotal in many applications from mathematical modeling to engineering where predicting behavior as inputs approach boundaries define system constraints and limits.

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Most popular questions from this chapter

At time \(t\) hours after taking the cough suppressant hydrocodone bitartrate, the amount, \(A,\) in mg, remaining in the body is given by \(A=10(0.82)^{t}.\) (a) What was the initial amount taken? (b) What percent of the drug leaves the body each hour? (c) How much of the drug is left in the body 6 hours after the dose is administered? (d) How long is it until only 1 mg of the drug remains in the body?

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Explain what is wrong with the statement. $$\ln (A+B)=(\ln A)(\ln B).$$
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