Problem 40 - Thomas Calculus

Question

Find the limits in Exercises \(37-48.\) $$\lim _{x \rightarrow 3^{+}} \frac{1}{x-3}$$

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Analyze the Function

Consider the function \( f(x) = \frac{1}{x-3} \) and note how it behaves as \( x \) approaches 3 from the right, or positively (\( x \to 3^+ \)). As \( x \to 3^+ \), the value of \( x-3 \) approaches 0 from the positive side, making the denominator very small.

Step 2: Determine the Limit Behavior

As \( x \to 3^+ \), the expression \( x-3 \) becomes a small positive number. Thus, \( \frac{1}{x-3} \) becomes a very large positive number because division by a small positive number results in a large positive quotient.

Step 3: Conclusion of Limit

Based on the above analysis, we conclude that as \( x \) approaches 3 from the right, \( \frac{1}{x-3} \) tends towards infinity. Therefore, the limit is \( \infty \).

Key Concepts

Asymptotic Behavior

Understanding asymptotic behavior involves examining how a function behaves as it gets close to a certain value or point. When we observe a function near a point where it cannot establish a standard value due to a denominator approaching zero or because it shoots off to infinity, we are talking about its asymptotic behavior. For the function \( f(x) = \frac{1}{x-3}\), we are interested in how it behaves asymptotically as we approach \( x=3 \) from the right.

When approaching the given point \( x=3 \) from positive values, the denominator \( x-3 \) gets smaller. As \( x \) closes in on this point without ever actually reaching it (since division by zero is undefined), the function value \( \frac{1}{x-3} \) rises dramatically. This vertical trend towards infinity as we near the point from one direction constitutes its asymptotic behavior, highlighting the point where it can no longer compute a meaningful or finite number.

One-Sided Limits

One-sided limits examine the behavior of a function as it approaches a certain point from only one side. In mathematical terms, these are the limits of a function as \( x \) approaches a particular value from either the left \( (x \to a^-) \) or the right \( (x \to a^+) \). These limits help us understand whether the function behaves differently depending on the direction from which we approach a specific point.

For the function \( \frac{1}{x-3} \), considering the limit as \( x \to 3^+ \) is an example of a one-sided limit. We seek to understand what happens to the function as \( x \) draws closer to 3 from the right side only. This concept is essential because it shows that indeed, as \( x \) nears 3 from positive values, the function shoots towards positive infinity. This analysis doesn't attempt from the left, where values would be negative, showcasing the unique one-sided behavior of functions near undefined points.

Infinite Limits

Infinite limits describe a situation where the function values grow endlessly large (positive or negative) as we approach a specific point. These limits explore the fun scenarios where once close, but still not quite there, the output of the function races off to either positive or negative infinity.

In the exercise involving \( \lim_{x \to 3^+} \frac{1}{x-3} \), we're dealing with an infinite limit. As \( x \to 3^+ \), the denominator \( x-3 \) inches towards zero from the positive side. Because the fraction's denominator is diminishing to a small positive number, the value of \( \frac{1}{x-3} \) becomes exceedingly large in a positive sense. Here, since the quotient scales larger without bound as \( x \) approaches 3, we conclude that the limit is infinite, meaning there is no finite number it settles at. Thus, we express this understanding in mathematical jargon by saying the limit tends to positive infinity, written as \( \lim_{x \to 3^+} \frac{1}{x-3} = \infty \). This serves as a prime example of how limits can communicate the concept of unbounded behavior.