Problem 60 - Thomas Calculus

Question

Find the limits in Exercises \(59-62\). $$\lim \left(\frac{1}{t^{3 / 5}}+7\right)\mathrm{as}$$ $$\text { a. }t \rightarrow 0^{+} \quad \text { b. } t \rightarrow 0^{-} $$

Step-by-Step Solution

Answer

a. \(+\infty\); b. \(-\infty\).

Step 1: Understand the Problem

The given problem requires us to find the limit of the expression \( \frac{1}{t^{3/5}} + 7 \) as \( t \) approaches 0 from the positive side (\( 0^+ \)) and from the negative side (\( 0^- \)). The expression involves a fractional exponent, which impacts how the expression behaves around zero.

Step 2: Examine Limit as t Approaches 0 from the Positive Side

As \( t \to 0^+ \), the term \( \frac{1}{t^{3/5}} \) becomes extremely large because the denominator \( t^{3/5} \) approaches zero, so the fraction approaches infinity. Therefore, the expression \( \frac{1}{t^{3/5}} + 7 \) also approaches infinity. The positive side means \( t \) is a small positive number, making the expression blow up to positive infinity. Hence, the limit is: \( \lim_{{t \to 0^+}} \left( \frac{1}{t^{3/5}} + 7 \right) = +\infty \).

Step 3: Examine Limit as t Approaches 0 from the Negative Side

As \( t \to 0^- \), the expression \( \frac{1}{t^{3/5}} \) becomes extremely large in magnitude, but since \( t^{3/5} \) has an odd power fractional exponent, the result is negative because \( t \) is negative. Thus, \( \frac{1}{t^{3/5}} \to -\infty \) and the entire expression \( \frac{1}{t^{3/5}} + 7 \to -\infty \) as \( t \to 0^- \). Therefore, \( \lim_{{t \to 0^-}} \left( \frac{1}{t^{3/5}} + 7 \right) = -\infty \).

Key Concepts

Fractional Exponents

Fractional exponents provide a unique perspective on how numbers behave, especially when comparing them to whole number exponents. A fractional exponent like \( t^{3/5} \) is essentially an extension of the square root and cube root concept. Here, the exponent \( 3/5 \) means take the cube of \( t \) first and then the fifth root of that result, or vice versa. When dealing with fractional exponents close to zero, the number \( t \) raised to an exponent smaller than 1 will crunch to a very small number if \( t \) itself is positive and close to zero. However, if \( t \) is negative, its behavior is particularly interesting. A fraction whose numerator and denominator are both odd (like \( 3/5 \)) will retain the sign of \( t \). Thus, if \( t \) is negative, raising it to the \( 3/5 \)-th power keeps it negative, multiplying it by a smaller number closer to zero but maintaining the negative sign.

• For positive \( t \), \( t^{3/5} \rightarrow 0^+ \) quickly as \( t \rightarrow 0^+ \).
• For negative \( t \), \( t^{3/5} \rightarrow 0^- \) quickly as \( t \rightarrow 0^- \), keeping negative sign.

These peculiarities impact limits heavily, as we'll see in one-sided limits.

One-Sided Limits

One-sided limits are a crucial concept in calculus, allowing us to analyze the behavior of functions as they approach a particular point from one direction—either from the right (\( 0^+ \)) or from the left (\( 0^- \)). To find the one-sided limit of \( \frac{1}{t^{3/5}} + 7 \) as \( t \rightarrow 0^+ \), observe that as the fractional exponent \( t^{3/5} \) approaches zero, the division \( \frac{1}{t^{3/5}} \) yields a very large positive value, driving the entire expression towards positive infinity.Analyzing from the left-hand direction, as \( t \rightarrow 0^- \), the fractional exponent \( t^{3/5} \) turns negative. Dividing 1 by a tiny negative number results in negative infinity, leading the expression to plummet towards negative infinity.To summarize these observations:

• Limit from the positive side: As \( t \rightarrow 0^+ \), \( \frac{1}{t^{3/5}} + 7 \rightarrow +\infty \).
• Limit from the negative side: As \( t \rightarrow 0^- \), \( \frac{1}{t^{3/5}} + 7 \rightarrow -\infty \).

Understanding one-sided limits helps distinguish the different behaviors that a function might exhibit when approaching the same point from different directions.

Asymptotic Behavior

Asymptotic behavior in calculus refers to the tendency of a function to approach a line but never quite reaching it, as it moves towards infinity or zero. This concept gives insight into how functions behave under extreme conditions.In the expression \( \frac{1}{t^{3/5}} + 7 \), as \( t \) approaches zero, the main impactful term is \( \frac{1}{t^{3/5}} \). This term embodies asymptotic behavior by becoming a dominant factor that skyrockets towards infinity—either positive or negative—based on the direction \( t \) is approaching from.

• When approaching zero from the positive side, \( t^{3/5} \rightarrow 0^+ \) makes \( \frac{1}{t^{3/5}} \rightarrow +\infty \). Thus, \( \frac{1}{t^{3/5}} + 7 \) also tends towards positive infinity.
• When approaching from the negative side, \( t^{3/5} \rightarrow 0^- \). Therefore, \( \frac{1}{t^{3/5}} \rightarrow -\infty \), causing \( \frac{1}{t^{3/5}} + 7 \) to drift to negative infinity.

These asymptotic tendencies essentially define how the function behaves near crucial points, enabling predictions of function values even without graphing them.