Problem 21 - Calculus: Early Transcendentals

Question

Determining limits analytically Determine the following limits. a. \(\lim _{x \rightarrow 2^{+}} \frac{1}{x-2}\) b. \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\) c. \(\lim _{x \rightarrow 2} \frac{1}{x-2}\)

Step-by-Step Solution

Answer

Question: Determine the limits of the function \(f(x) = \frac{1}{x-2}\) for a) \(x\) approaching 2 from the right, b) \(x\) approaching 2 from the left, and c) \(x\) approaching 2 in general. Answer: a) \(+\infty\), b) \(-\infty\), and c) does not exist.

Step 1: Understand the direction

In this case, we are considering the limit as \(x\) approaches \(2\) from the right side (indicated by \(2^+\)), which means \(x\) will be slightly greater than 2.

Step 2: Use the fact that \(x\) is slightly greater than 2

Since \(x\) is slightly greater than 2, \((x-2)\) will be positive. This means that \(\frac{1}{x-2}\) will also be positive.

Step 3: Determine the limit

As \(x\) gets closer to 2 from the right side, the value of \((x-2)\) gets closer to 0 (but always positive). So, the value of \(\frac{1}{x-2}\) gets larger and larger. Therefore, \(\lim_{x \rightarrow 2^+} \frac{1}{x-2} = +\infty\). b. \(\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}\)

Step 1: Understand the direction

In this case, we are considering the limit as \(x\) approaches \(2\) from the left side (indicated by \(2^-\)), which means \(x\) will be slightly smaller than 2.

Step 2: Use the fact that \(x\) is slightly smaller than 2

Since \(x\) is slightly smaller than 2, \((x-2)\) will be negative. This means that \(\frac{1}{x-2}\) will also be negative.

Step 3: Determine the limit

As \(x\) gets closer to 2 from the left side, the value of \((x-2)\) gets closer to 0 (but always negative). So, the value of \(\frac{1}{x-2}\) gets larger and larger into the negative direction. Therefore, \(\lim_{x \rightarrow 2^-} \frac{1}{x-2} = -\infty\). c. \(\lim _{x \rightarrow 2} \frac{1}{x-2}\)

Step 1: Understand the limit

We want to determine the general limit as \(x\) approaches \(2\). For this limit to exist, both the left and right-sided limits must exist and be equal.

Step 2: Compare the left and right-sided limits

From parts a and b, we found that the left-sided limit is \(-\infty\) and the right-sided limit is \(+\infty\). These values are not equal.

Step 3: Conclude the general limit

Since the left-sided and right-sided limits are not equal, the general limit does not exist. Therefore, \(\lim_{x \rightarrow 2} \frac{1}{x-2}\) does not exist.

Key Concepts

One-Sided Limits

When you're dealing with limits in calculus, one-sided limits are an important concept. They help us understand what happens to a function as it approaches a certain point from one specific direction, either from the left or right.

• Right-Sided Limit: This is denoted as \(\lim_{x \rightarrow c^{+}} f(x)\). It represents the value that the function approaches as \(x\) gets close to \(c\) from the right side (values greater than \(c\)). For example, in our exercise, \(\lim_{x \rightarrow 2^{+}} \frac{1}{x-2} = +\infty\) means that as \(x\) comes from values slightly greater than 2, the function's value becomes infinitely large.
• Left-Sided Limit: This is expressed as \(\lim_{x \rightarrow c^{-}} f(x)\). It describes the value the function takes when \(x\) approaches \(c\) from the left side (values less than \(c\)). For instance, in our example, \(\lim_{x \rightarrow 2^{-}} \frac{1}{x-2} = -\infty\), which means as \(x\) nears 2 from the left, the function values drop toward negative infinity.

Understanding one-sided limits is crucial for grasping the concept of continuity and the behavior of functions at specific points.

Infinite Limits

Infinite limits describe situations where a function approaches infinity (positive or negative) as \(x\) gets closer to a certain value.

• Positive Infinity: If \(\lim_{x \rightarrow c} f(x) = +\infty\), the function values become larger and larger without bound as \(x\) approaches \(c\). In part (a) of our exercise, as \(x\) gets closer to 2 from the right, \(\frac{1}{x-2}\) increases towards positive infinity.
• Negative Infinity: Conversely, \(\lim_{x \rightarrow c} f(x) = -\infty\) means the function values decrease without limit towards negative infinity. In part (b), as \(x\) approaches 2 from the left, \(\frac{1}{x-2}\) goes down to negative infinity.

Infinite limits are not finite, but they indicate the direction of growth or decay of a function. They are useful for understanding vertical asymptotes and behavior at boundaries.

Limit Does Not Exist

Sometimes, limits simply do not exist. This happens when the function does not approach a single value as \(x\) approaches the point of interest.

• Different One-Sided Limits: If the left-sided limit and right-sided limit are not equal, the general limit at that point does not exist. For example, in part (c) of our exercise, \(\lim_{x \rightarrow 2} \frac{1}{x-2}\) does not exist because the left-sided limit is \(-\infty\), and the right-sided limit is \(+\infty\).
• Oscillation: Another reason a limit might not exist is if the function oscillates between values as it approaches the point, rather than settling to a single value.

Understanding when a limit does not exist is crucial for analyzing discontinuities and making sense of a function's overall behavior.