Problem 47 - Precalculus

Question

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow-1} \frac{1}{(x+1)^{2}}$$

Step-by-Step Solution

Answer

The limit as \(x\) approaches \(-1\) does not exist because the function tends to infinity.

Step 1: Identify the Expression

We are given the function \(f(x) = \frac{1}{(x+1)^{2}} \) and need to find the limit as \( x \) approaches \(-1\). This will involve looking at both left-hand and right-hand limits to see if they agree.

Step 2: Analyze Left-Hand Limit

To find the left-hand limit as \( x \) approaches \(-1\), consider values of \( x \) that are slightly less than \(-1\). As \( x \to -1^{-}\), the function \( \frac{1}{(x+1)^{2}} \) tends towards \( \infty \) because \((x+1)^2\) becomes a very small positive number, making the fraction very large.

Step 3: Analyze Right-Hand Limit

To find the right-hand limit as \( x \) approaches \(-1\), consider values of \( x \) that are slightly greater than \(-1\). Similarly, as \( x \to -1^{+}\), \( \frac{1}{(x+1)^{2}} \) also tends towards \( \infty \). This is because the square makes it positive, and any small number close to zero in the denominator makes the fraction very large.

Step 4: Determine Overall Limit

Since both the left-hand and right-hand limits as \( x \to -1 \) result in the function approaching \( \infty \), we conclude that the limit does not exist in the traditional sense. Although both sides go to infinity, the limit is not a real number.

Step 5: Conclusion and Explanation

While both sides of the limit approach \( \infty \), the limit as \( x \to -1 \) of \( \frac{1}{(x+1)^{2}} \) does not exist because the result is not a finite number. Instead, we say the function increases without bound near \( x = -1 \).

Key Concepts

Graphing Utility

When dealing with calculus problems that involve limits, a graphing utility can be a powerful tool. Graphing utilities, like graphing calculators or software, allow you to visualize the behavior of functions near specific points. For this exercise, you can use a graphing utility to plot the function \(f(x) = \frac{1}{(x+1)^{2}}\).

By entering the function into the graphing utility and observing its behavior as \(x\) approaches \(-1\), you can see how the function increases rapidly. In this case, both sides of the function appear to shoot upwards as \(x\) gets close to \(-1\).

Using a graphing utility helps in analyzing the nature of limits because it provides a visual representation of how the function behaves, making it easier to determine whether the function approaches a specific value or diverges.

Left-Hand Limit

In calculus, the left-hand limit refers to the value that a function approaches as the input, \(x\), approaches a specific point from the left side. In this problem, we are examining what happens to \(f(x) = \frac{1}{(x+1)^{2}}\) as \(x\) approaches \(-1\) from values slightly less than \(-1\), noted as \(x\to -1^{-}\).

For the given function, as \(x\) gets closer to \(-1\) from the left, \(x + 1\) becomes a very small negative number, but squaring it results in a small positive number. Hence, \(\frac{1}{(x+1)^2}\) becomes very large. As a result, the left-hand limit of the function is \(+\infty\).

• Left-hand limit explores function behavior as \(x\) nears a specific point from the left.
• Helpful to determine if both sides of the limit at the point converge.

Right-Hand Limit

The right-hand limit is the value a function approaches as the input, \(x\), approaches a specific point from the right. For \(f(x) = \frac{1}{(x+1)^{2}}\), we are interested in what happens as \(x\) gets close to \(-1\) from values slightly greater than \(-1\), indicated as \(x\to -1^{+}\).

As \(x\) inches towards \(-1\) from the right, \(x + 1\) becomes a small positive number near zero. Again, squaring this small number yields another small positive number, resulting in the overall fraction \(\frac{1}{(x+1)^2}\) growing very large. Thus, the right-hand limit is also \(+\infty\).

• Right-hand limit studies the function's behavior as \(x\) approaches from the right.
• Essential for comparing with the left-hand limit to determine if both are the same.

Infinite Limit

An infinite limit describes a situation where a function grows without bound as \(x\) approaches a certain point. In this exercise, both the left-hand and right-hand limits of \(f(x) = \frac{1}{(x+1)^{2}}\) as \(x\to -1\) result in \(+\infty\).

Although both sides approaching \(\infty\) might suggest consistency, an infinite limit means the limit does not exist in the conventional sense. Limits that result in infinity indicate that the function doesn't settle at a particular finite value.

• Infinite limits show a function's tendency to grow without constraint near a point.
• Indicates that traditional limits are undefined for a specific function behavior.

Understanding infinite limits is key in diagnosing how functions behave at points that cause unbounded growth.