Problem 57 - Precalculus with Limits

Question

In Exercises 55-62, graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. $$\lim_{x \to 1} \dfrac{1}{x^2+1}$$

Step-by-Step Solution

Answer

The short answer will be given by the common value found in step 4, if such value exists. If the value doesn't exist, the limit doesn't exist either.

Step 1: Plotting the Graph

Plot the function \(f(x) = \dfrac{1}{x^2+1}\) using a graphing tool. This will help in visualizing how the function behaves as x approaches 1.

Step 2: Calculation of the Left Limit

Calculate the left-hand limit of \(f(x) = \dfrac{1}{x^2+1}\) as x approaches 1. It is denoted by \(\lim_{x \to 1^-} \dfrac{1}{x^2+1}\) and represents how the function behaves as x approaches 1 from the left on the number line.

Step 3: Calculation of the Right Limit

Similarly, calculate the right-hand limit of \(f(x) = \dfrac{1}{x^2+1}\) as x approaches 1. Denoted by \(\lim_{x \to 1^+} \dfrac{1}{x^2+1}\), this displays how the function behaves as x approaches 1 from the right.

Step 4: Determining the Limit

If step 2 and step 3 yield the same value, it means the function value approaches that common value as x approaches 1 from both sides. This value is then the limit of function as x approaches 1, i.e., \(\lim_{x \to 1} \dfrac{1}{x^2+1} =\) that common value. Otherwise, if the one-sided limits are unequal, the limit of function as x approaches 1 does not exist.

Key Concepts

One-Sided Limits

In calculus, one-sided limits are used to understand the behavior of a function as it approaches a specific value from one side only, either from the left or the right. These are crucial when the function behaves differently on either side of the point.
- **Left-Hand Limit (\(\lim_{x \to 1^-} \frac{1}{x^2+1}\))**: This describes the behavior of the function as x approaches 1 from values less than 1.
- **Right-Hand Limit (\(\lim_{x \to 1^+} \frac{1}{x^2+1}\))**: This shows how the function behaves as x approaches 1 from values greater than 1.For the limit to exist at a certain point, both the left-hand and right-hand limits must be equal. If they differ, the function does not have a limit at that point. This concept helps us pinpoint discontinuities or sudden jumps in a function.

Graphing Functions

Graphing a function is a visual way to examine its behavior. It provides a comprehensive picture of how a function behaves across different values of x. In exercises like this, graphing becomes an indispensable tool to easily visualize \( f(x) = \frac{1}{x^2+1} \).
- **Identify Key Features**: When graphing functions, look for intercepts, asymptotes, and specific intervals of interest. This helps in understanding where to focus as x approaches a point.
- **Behavior Around the Point**: Particularly for limits, observing how a function behaves around x = 1 can help confirm the calculated value of the limit. You can visually check if the graph approaches a single, consistent value from both sides.
Employing graphing tools or graphing the function by hand enhances understanding and supports the calculations made during the limit evaluation process.

Calculus

Calculus is the branch of mathematics that studies continuous change, encapsulating concepts like limits, derivatives, and integrals. When evaluating limits, like the exercise above, you are engaging in one of calculus' fundamental activities.
- **Limits**: A key concept in calculus, limits help us understand the behavior of functions as they approach a certain point. They are foundational for defining derivatives and integrals.
- **Practical Application**: Even concepts like one-sided limits and graphing provide insight into more complex calculus operations. They are often used to solve problems involving rates of change and areas under curves.
Understanding these basic elements of calculus will definitely enhance your ability to tackle similar problems, bridge theoretical understanding with practical analysis, and prepare for more advanced studies in calculus topics.