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Chapter 2: Problem 15

Find the limits. $$\lim _{x \rightarrow 2} \frac{2 x+5}{11-x^{3}}$$

Short Answer

Expert verified
The limit is 3.

Step by step solution

01

Understand the Expression

We need to find the limit of \( \frac{2x+5}{11-x^3} \) as \( x \) approaches 2. This means substituting \( x \) with 2 in the expression to see if it leads to a determinate or indeterminate form.
02

Substitute x with 2

Substitute \( x = 2 \) directly into the expression: \( \frac{2(2) + 5}{11 - (2)^3} \). Simplifying, we get \( \frac{4 + 5}{11 - 8} \). This simplifies further to \( \frac{9}{3} \).
03

Simplify the Result

The fraction \( \frac{9}{3} \) simplifies to 3, as dividing 9 by 3 gives us 3.
04

Determine the Limit

Since substituting \( x = 2 \) into the expression gave a valid numerical result without any division by zero or undefined operations, the limit exists and equals the simplified result from Step 3.

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Key Concepts

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

Indeterminate Forms
When calculating the limit of a function, you may encounter expressions that can鈥檛 be directly evaluated because they result in indeterminate forms. These forms include \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), and others, which do not provide enough information to determine the actual limit of the function.
  • An indeterminate form often signifies that direct substitution into the limit expression won鈥檛 work straightforwardly and further simplification is required.
  • In our given exercise, substituting \(x = 2\) into \(\frac{2x+5}{11-x^3}\) directly yields a determinate form because it doesn鈥檛 lead to division by zero or undefined operations.
Understanding indeterminate forms is crucial because it directs the approach towards either algebraic manipulation or employing special theorems and rules, such as L'H么pital's Rule, for more complex cases.
Substitution Method
The substitution method is a straightforward technique in finding limits where you directly replace the variable with the given value. This helps check whether the resulting expression is determinate or indeterminate.
  • Start by directly substituting the specified value of \(x\) into the expression.
  • If simplification leads to a clear numerical value without encountering an indeterminate form, you have found the limit using this method.
For the problem \(\lim_{x \rightarrow 2} \frac{2x+5}{11-x^3}\), substitution gives \(\frac{4 + 5}{11 - 8} = \frac{9}{3}\), leading to the determinate result of 3. The method worked here efficiently because the operation was straightforward, showcasing when substitution is effective.
Simplifying Fractions
Simplifying fractions is a basic but essential step in mathematical evaluations, especially in limits, to reach the final compact form of the expression.
  • After substituting and calculating, you may often end up with a fraction that can be reduced to its simplest form, enhancing clarity and understanding.
  • Dividing both the numerator and the denominator by their greatest common divisor is critical.
In the provided limit problem, after substitution, we calculated \( \frac{9}{3} \), which simplifies to 3. Breaking down larger or complex fractions into a simplified form helps in making quick and accurate decisions about limits and their evaluations. Having mastery over simplifying fractions not only makes limit problems easier but also helps in various branches of mathematics.

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Most popular questions from this chapter

Graph the function \(f\) to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at \(x=0 .\) If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be? $$f(x)=\frac{\sin x}{|x|}$$

Show that the function \(F(x)=(x-a)^{2}\) \((x-b)^{2}+x\) takes on the value \((a+b) / 2\) for some value of \(x\)

Prove the limit statements. $$\lim _{x \rightarrow-2} f(x)=4 \quad \text { if } \quad f(x)=\left\\{\begin{array}{ll}x^{2}, & x \neq-2 \\\1, & x=-2\end{array}\right.$$

You will find a graphing calculator useful for Exercise. Let \(G(t)=(1-\cos t) / t^{2}.\) a. Make tables of values of \(G\) at values of \(t\) that approach \(t_{0}=0\) from above and below. Then estimate \(\lim _{t \rightarrow 0} G(t).\) b. Support your conclusion in part (a) by graphing \(G\) near \(t_{0}=0.\)

$$\text { Prove that } \lim _{x \rightarrow c} f(x)=L \text { if and only if } \lim _{h \rightarrow 0} f(h+c)=L$$
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