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Chapter 12: Problem 38

Evaluating One-Sided Limits graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. $$ \lim _{x \rightarrow 1} \frac{1}{x^{2}+1} $$

Short Answer

Expert verified
\(\lim _{x \rightarrow 1} \frac{1}{x^{2}+1} = 0.500\).

Step by step solution

01

One-sided limit from the left

To evaluate the one-sided limit from the left (\(x\) approaches 1 from values less than 1), we substitute \(x\) with a value slightly less than 1 into the function. For instance \(x = 0.999\). \(\lim _{x \rightarrow 1^{-}} \frac{1}{x^{2}+1} = \frac{1}{(0.999)^{2}+1} = 0.500\). We can see here the function is approaching the value \(0.500\) as \(x\) approaches 1 from the left.
02

One-sided limit from the right

To evaluate the one-sided limit from the right (\(x\) approaches 1 from values greater than 1), we substitute \(x\) with a value slightly greater than 1 into the function. For example \(x = 1.001\). \(\lim _{x \rightarrow 1^{+}} \frac{1}{x^{2}+1} = \frac{1}{(1.001)^{2}+1} = 0.500\). Here we see that the function is also approaching the value \(0.500\) as \(x\) approaches 1 from the right.
03

Evaluating the Limit

Since the one-sided limits from both the left and the right are equal at 0.500, that means the overall limit exists and is equal to 0.500. Thus, \(\lim _{x \rightarrow 1} \frac{1}{x^{2}+1} = 0.500\).

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Key Concepts

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

Limit Evaluation
When evaluating limits, especially one-sided limits, it鈥檚 important to consider the approach from both the left and the right of a particular point. This is essential in understanding if a limit exists as you approach that point.

Here is how you tackle a one-sided limit evaluation:
  • **One-sided limits** involve analyzing the behavior of a function as it approaches a specific point from just one side (either the left or right). In the example given, this includes evaluating \( \lim_{x \to 1^{-}} \) and \( \lim_{x \to 1^{+}} \).
  • Accurate **substitution** is required by choosing values very close to the point of interest from either side. For example, substituting values like \( x = 0.999 \) and \( x = 1.001 \) helps estimate the behavior of \( \frac{1}{x^2 + 1} \) as \( x \) approaches 1.
  • If both one-sided limits are equal, then the general limit, \( \lim_{x \to 1} \), exists and is equal to that common value.
Understanding one-sided limits highlights how a function behaves around a particular point, ensuring precise approximation.
Graphical Analysis
Graphical analysis is another powerful method to understand limits, including one-sided limits. By visualizing the function on a graph, you can plainly see how it behaves as it approaches specific values.

Consider graphing the function \( \frac{1}{x^2+1} \):
  • **Plotting the function** helps visualize both directions as \( x \) closes in on your point of interest, in this scenario, as \( x \) approaches 1.
  • By identifying the trend lines, we can see how values converge to a specific number as we get closer to \( x = 1 \) from both sides.
  • Visual indications on the graph help confirm if a function consistently nears a particular limit from different directions, in this case, matching the result found in the step-by-step evaluation.
Graphical representation provides a clear visual aid in verifying findings from algebraic evaluations and magnifies your comprehension of the concept of limits.
Calculus Concepts
In calculus, understanding limits is foundational to grasping more advanced concepts like derivatives and integrals. Let鈥檚 delve into why one-sided limits are pivotal:
  • **Continuity:** One-sided limits help determine whether a function is continuous at a point. For a function to be continuous at a point, the one-sided limits must be the same, which means the overall limit exists at that point.
  • **Differentiability:** If functions differ in their one-sided limits, the function may not be differentiable at that point. Evaluating these limits cautions us about points of discontinuity or sharp turns.
  • **Smooth Analysis:** By investigating limits deeply, calculus allows for smoother analysis of functions, providing the ability to predict behavior beyond static calculations.
Knowing how to evaluate one-sided limits fundamentally aids in understanding critical points like continuity and differentiability in calculus, offering insights into more advanced topics.

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Most popular questions from this chapter

The path of a ball thrown by a child is modeled by \(y=-x^{2}+5 x+2\) where \(y\) is the height of the ball (in feet) and \(x\) is the horizontal distance (in feet) from the point from which the child threw the ball. Using your knowledge of the slopes of tangent lines, show that the height of the ball is increasing on the interval \([0,2]\) and decreasing on the interval \([3,5] .\) Explain your reasoning.

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. Function \(\quad\) Line \(f(x)=-\frac{1}{2} x^{3} \quad 6 x+y+4=0\)

Average cost The cost function for a certain model of MP3 player is given by \(C = 73 x + 25,000 ,\) where \(C\) is in dollars and \(x\) is the number of MP3 players produced. (a) Write a model for the average cost per unit produced. (b) Find the average costs per unit when \(x = 1000\) and \(x = 5000\) . (c) Determine the limit of the average cost function as \(x\) approaches infinity. Explain the meaning of the limit in the context of the problem.

Finding the Area of a Region, use the limit process to find the area of the region bounded by the graph of the function and the \(x\) -axis over the specified interval. $$\begin{array}{ll}{\text { Function }} & {\text { Interval }} \\ {f(x)=2 -x^{2}} & {[-1,1]}\end{array}$$

Finding the Area of a Region,complete the table to show the approximate area of the region bounded by the graph of \(f\) and the \(x\) -axis over the specified interval using the indicated numbers \(n\) of rectangles of equal width. Then find the exact area as \(n \rightarrow \infty\). $$\begin{array}{ll}{\text { Function }} & {\text { Interval }} \\ {f(x)=x^{2} + {1}} & {[4,6]}\end{array}$$
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