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Magazine Chapter 2: Problem 29
Find the limits. $$ \lim _{t \rightarrow 3^{-}} \frac{t^{2}}{9-t^{2}} $$
Short Answer
Expert verified
The limit is positive infinity.
Step by step solution
01
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.
02
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.
03
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.
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
- This analysis is crucial because the denominator approaching zero indicates that the function's value will grow significantly, given any non-zero numerator.
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.
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.
