Problem 23 Find the limits in Exercises \(2... [FREE SOLUTION]

Chapter 2: Problem 23

Find the limits in Exercises $23-26$. $$ \begin{array}{l}{\lim \left(2-\frac{3}{t^{1 / 3}}\right) \text { as }} \\\ {\text { a. } t \rightarrow 0^{+} \quad \text { b. } t \rightarrow 0^{-}}\end{array} $$

Short Answer

Expert verified

$\lim_{t \to 0^+} \left(2-\frac{3}{t^{1/3}}\right) = -\infty$ and $\lim_{t \to 0^-} \left(2-\frac{3}{t^{1/3}}\right) = +\infty$.

Step by step solution

Understand the Problem

We are given the expression $2 - \frac{3}{t^{1/3}}$ and need to find the limit as $t$ approaches zero from the positive side ($t \to 0^+$) and from the negative side ($t \to 0^-$). We analyze the behavior of $\frac{3}{t^{1/3}}$ under these conditions.

Analyze $t \to 0^+$

For $t \to 0^+$, $t$ is a very small positive number. Consequently, $t^{1/3}$ is also a small positive number, and $\frac{3}{t^{1/3}}$ becomes very large and positive. Therefore, the expression $2 - \frac{3}{t^{1/3}}$ becomes $2 - \text{(large positive number)}$, which tends towards $-\infty$.

Analyze $t \to 0^-$

For $t \to 0^-$, $t$ is a small negative number. $t^{1/3}$ becomes a small negative number because the cube root of a negative number remains negative. Hence, $\frac{3}{t^{1/3}}$ is a large negative number, making the expression $2 - \text{(large negative number)}$ result in a very large positive number. This means the expression tends towards $+\infty$.

Conclusion

Combining both analyses:- As $t \to 0^+$, the limit is $-\infty$.- As $t \to 0^-$, the limit is $+\infty$.

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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 simplify understanding how a function behaves as we approach a point from one direction. For instance, when dealing with the function $2 - \frac{3}{t^{1/3}}$, we're particularly interested in seeing how this function behaves as $t$ approaches zero from the right (positive side, $t \to 0^+$) and from the left (negative side, $t \to 0^-$).

Right-Hand Limit ($t \to 0^+$): Here, $t$ is getting close to zero, but is positive. $t^{1/3}$ will also be small and positive, making $\frac{3}{t^{1/3}}$ a large positive number. So, $2 - \frac{3}{t^{1/3}}$ tends towards $-\infty$.
Left-Hand Limit ($t \to 0^-$): In this case, $t$ approaches zero from the negative direction. $t^{1/3}$ is negative and small, hence $\frac{3}{t^{1/3}}$ becomes a large negative number. This makes $2 - \frac{3}{t^{1/3}}$ head towards $+\infty$.

Understanding these one-sided limits helps predict the direction in which a function heads as it near a point from specific paths.

Behavior of Functions

The behavior of functions, especially as they approach a specific point, reveals important insights into how that function behaves broadly. The function $2 - \frac{3}{t^{1/3}}$ clearly shows different behaviors when $t$ is near zero, depending on whether $t$ is positive or negative.

When $t$ is positive and close to zero, the behavior of $\frac{3}{t^{1/3}}$ causes the entire expression to decrease rapidly, resulting in large negative values ($-\infty$).
With a negative $t$ near zero, this changes dramatically. $\frac{3}{t^{1/3}}$, being negative too, will subtract a large negative number (which effectively adds a large number), pushing the function far into positive values ($+\infty$).

Recognizing how different values of $t$ affect the function helps build a more comprehensive understanding of its overall behavior.

Limits at Zero

Limits at zero can often indicate significant features about the continuity and performance of a function at and near this crucial point. For the expression $2 - \frac{3}{t^{1/3}}$, examining the limit at zero from both sides reveals differing results.

From the positive side ($t \to 0^+$), the limit tends towards infinity in the negative direction. This suggests that just before reaching zero, the function drops off dramatically.
From the negative side ($t \to 0^-$), we're observing towards $+\infty$. This captures a rapid upward shoot just as $t$ crosses zero from the negative spectrum.

The asymmetrical nature of these behaviors represents a discontinuity at $t = 0$. This indicates that the function doesn't settle into a single predictable value, but rather divides sharply between two extreme directions.

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91影视

Find the limits in Exercises \(23-26\). $$ \begin{array}{l}{\lim \left(2-\frac{3}{t^{1 / 3}}\right) \text { as }} \\\ {\text { a. } t \rightarrow 0^{+} \quad \text { b. } t \rightarrow 0^{-}}\end{array} $$

Short Answer

Step by step solution

Understand the Problem

Analyze \(t \to 0^+\)

Analyze \(t \to 0^-\)

Conclusion

Key Concepts

One-Sided Limits

Behavior of Functions

Limits at Zero

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