Problem 22 - Calculus: Early Transcendentals

Question

Determining limits analytically Determine the following limits. a. \(\lim _{x \rightarrow 3^{+}} \frac{2}{(x-3)^{3}}\) b. \(\lim _{x \rightarrow 3^{-}} \frac{2}{(x-3)^{3}} \quad\) c. \(\lim _{x \rightarrow 3} \frac{2}{(x-3)^{3}}\)

Step-by-Step Solution

Answer

" Answer: As x approaches 3 from the right, the limit is \(+\infty\). As x approaches 3 from the left, the limit is \(-\infty\). Since the right-hand and left-hand limits are different, the overall limit at x = 3 does not exist.

Step 1: Approach from the right

As we are looking for the right-hand limit, we need to analyze the function behavior as x approaches 3 from values greater than 3.

Step 2: Substitute values and analyze

As x approaches 3 from the right, the value (x-3) becomes very close to 0 but is a positive number. Since it is raised to the power of 3, the result will be very small and positive. The denominator will also be positive. Therefore, the expression becomes \(\frac{2}{small \thinspace positive}\). Thus, the limit is: a. \(\lim _{x \rightarrow 3^{+}} \frac{2}{(x-3)^{3}} = +\infty\) #b. Left-hand limit#

Step 1: Approach from the left

Since we are looking for the left-hand limit, we need to analyze the function behavior as x approaches 3 from values lower than 3.

Step 2: Substitute values and analyze

As x approaches 3 from the left, the value (x-3) becomes very close to 0 but is a negative number. Since it is raised to the power of 3, the result will be very small and negative. The denominator will be negative. Therefore, the expression becomes \(\frac{2}{small \thinspace negative}\). Thus, the limit is: b. \(\lim _{x \rightarrow 3^{-}} \frac{2}{(x-3)^{3}} = -\infty\) #c. Overall limit#

Step 1: Determine the overall limit

To determine the overall limit, we must look at the right-hand and left-hand limits. If they are different, the overall limit does not exist. c. As the right-hand limit is +∞ and the left-hand limit is −∞, the overall limit does not exist. Therefore, the limit is: c. \(\lim _{x\rightarrow 3} \frac{2}{(x-3)^{3}}\) does not exist.

Key Concepts

Right-Hand Limit

In calculus, the right-hand limit considers how a function behaves as the input approaches a certain value from the right side. Imagine a number line where you move closer to the target from higher values. In our example, we're examining the function \( \lim _{x \rightarrow 3^{+}} \frac{2}{(x-3)^{3}} \).

• As \( x \) approaches 3 from the right (values larger than 3), \( x - 3 \) is positive but very small, close to zero.
• Raising a positive small number to the third power still gives a positive result.
• The fraction \( \frac{2}{(x-3)^{3}} \) thus involves dividing 2 by a very small positive number, resulting in a very large positive number, or \(+\infty\).
  

Understanding this is critical for recognizing how functions can spike towards infinity from one side, defined specifically through right-hand limits.

Left-Hand Limit

The left-hand limit examines what happens to a function as you approach a specific input from smaller values. It's like sneaking up on that number from the left on a number line. For our exercise, consider \( \lim _{x \rightarrow 3^{-}} \frac{2}{(x-3)^{3}} \).

• When \( x \) is slightly less than 3, \( x-3 \) is negative and small, moving close to zero.
• Raising a small negative number to the power of three yields a negative value.
• Here, \( \frac{2}{(x-3)^{3}} \) calls for 2 divided by a tiny negative number, producing \(-\infty\).

It's important to grasp that coming from the left can drastically alter the limit's result compared to approaching from the right. Such differences play pivotal roles in understanding a function's one-sided behavior.

One-Sided Limits

One-sided limits focus on the behavior of functions from a single direction: either right or left but not both at once.

• The right-hand limit evaluates from above the target \( x \), indicating what happens with slightly larger values.
• The left-hand limit assesses from below, considering slightly smaller values than the target.

One-sided limits are essential for indicating how functions behave near certain points where they may not be well-behaved or continuous, like jumps or vertical asymptotes. In cases where the right-hand and left-hand limits differ, they can signal that the overall limit does not exist, as seen in our exercise where the limits yielded \(+\infty\) and \(-\infty\), respectively.

Infinity in Limits

Infinity often appears in limits, especially when a function increases or decreases without bound as it approaches a specific point. In our scenario, the limits of the function are \(+\infty\) and \(-\infty\).

• \(+\infty\) suggests that as you make very small positive movements within the function, the function's value grows infinitely large.
• \(-\infty\) indicates that similar negative movements cause the function to drop infinitely low.

The concept of being infinite isn't a number but expresses unbounded growth or decline. When limits like these diverge to \(+\infty\) or \(-\infty\), and especially when they conflict from different directions, as with our example, the general limit may not exist at that point. Infinity in limits reflects extreme behavior near particular inputs and is crucial in fully grasping and predicting functions' behaviors in calculus.