Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{t \rightarrow 3^{-}} \frac{t^{2}}{9-t^{2}} $$

Step-by-Step Solution

Answer

The limit is positive infinity.

Step 1: Understanding the Limit

We are tasked with finding the one-sided limit as \( t \) approaches 3 from the left (denoted as \( t \to 3^- \)) for the function \( f(t) = \frac{t^2}{9 - t^2} \). This means we need to consider values of \( t \) that are slightly less than 3.

Step 2: Observing the Denominator

Notice that as \( t \) approaches 3 from the left, the expression \( 9 - t^2 \) approaches zero because \( t^2 \) approaches 9. Since we are considering values slightly less than 3, \( t^2 \) will be slightly less than 9, making \( 9 - t^2 \) a small positive number.

Step 3: Analyzing the Numerator

The numerator \( t^2 \) approaches 9 as \( t \) approaches 3 from the left. Thus, the value of \( t^2 \) is close to 9 but never more than that when approaching from the left.

Step 4: Applying Limits to the Expression

As \( t \to 3^- \), both the numerator approaches 9 and the denominator approaches a very small positive number. The function \( \frac{t^2}{9 - t^2} \) therefore behaves like \( \frac{9}{0^+} \), which indicates the function tends towards positive infinity.

Key Concepts

One-Sided Limits

One-sided limits are a pivotal concept in calculus, allowing us to understand the behavior of a function as it approaches a particular point from one specific direction.
This is especially useful when the function has a distinct behavior on either side of the point. In this exercise, we are focusing on a one-sided limit, denoted as \( t \to 3^- \), meaning we examine the function as \( t \) approaches 3 but stays less than 3. This direction-focused analysis is crucial when examining functions with potential discontinuities or undefined points.

Approaching from the Left

When we talk about approaching a point from the left in calculus, we refer to values that are less than (but very close to) the target point. In our scenario, \( t \to 3^- \) means considering values of \( t \) slightly less than 3.

• These values mean we consider a limit that examines the "left-hand side" behavior of the function.


• Approaching from the left ensures that we only consider values where the directionality matters, such as the function \( f(t) \) approaching a specific value.

This directional approach provides insights that might be hidden when considering only the general behavior of a function.

Numerator and Denominator Analysis

Analyzing both the numerator and the denominator individually is essential in determining the behavior of a function near a point of interest.
In our exercise, the numerator \( t^2 \) was approaching 9.

• As \( t \to 3^- \), \( t^2 \) becomes slightly less than 9, reflecting the proximity to the square of 3.

Meanwhile, the denominator \( 9 - t^2 \) becomes a very small positive number due to \( t^2 \) nearing 9 from below.

• This analysis is crucial because the denominator approaching zero indicates that the function's value will grow significantly, given any non-zero numerator.

Being systematic with this breakdown sheds light on the significant influence denominators have on the overall behavior of the fraction.

Infinite Limits

Infinite limits reveal the behavior of functions that increase or decrease without bound as they approach a specific point.
In the given exercise, as \( t \to 3^- \), \( \frac{t^2}{9-t^2} \) tends towards \( \frac{9}{0^+} \), which is interpreted as positive infinity.

• This occurs because the numerator (approaching 9) remains finite, while the denominator becomes an exceedingly small positive number.


• Such tiny positive denominators make the fraction's value soar to an infinitely large positive number.

Understanding infinite limits helps explain how sometimes functions in calculus aren't bound by finite numbers, but rather exhibit extreme behavior towards positive or negative infinity.