Problem 37 - Thomas Calculus

Question

Find the limits in Exercises \(37-48.\) $$\lim _{x \rightarrow 0^{+}} \frac{1}{3 x}$$

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Understand the Limit Notation

We are asked to find the limit as \( x \) approaches 0 from the positive direction (right-hand limit). This is denoted as \( x \to 0^+ \).

Step 2: Analyze the Expression \( \frac{1}{3x} \)

The expression \( \frac{1}{3x} \) represents a function that becomes large as \( x \) becomes small. Specifically, as \( x \to 0^+ \), the denominator \( 3x \) approaches zero, and the fraction becomes very large.

Step 3: Determine Behavior as \( x \to 0^+ \)

Since \( x \) is positive and approaching zero from the right, \( 3x \) is a small positive number. As the denominator gets smaller (but positive), \( \frac{1}{3x} \) increases without bound. Therefore, the limit approaches infinity.

Key Concepts

Right-Hand Limit

A "right-hand limit" is a type of limit that examines the behavior of a function as the input value approaches a certain point from the right. In mathematical terms, this means considering what happens to a function as the variable \( x \) approaches a specific value from values greater than it or the positive side. We denote this situation using the notation \( x \to a^+ \), where \( a \) is the point of interest. For example, if \( x \to 0^+ \), it means we are seeing what happens when \( x \) approaches zero from right of zero or positive values close to zero.

• For example, the limit \( \lim_{x \to 0^+} \frac{1}{3x} \) focuses on values of \( x \) that are just slightly greater than zero.
• This is useful in determining the behavior of functions near points where they might not be defined or are discontinuous.

Understanding right-hand limits helps in comprehending how functions behave in boundary situations, which can be crucial in calculus studies and real-world applications.

Infinite Limit

An "infinite limit" occurs when a function grows indefinitely as the input approaches a certain point. In the context of calculus, if as \( x \to a \), the function value gets larger and larger (positively or negatively), we say that the function approaches infinity.
Let's consider the function \( \frac{1}{3x} \). As \( x \) nears zero from the positive side, the expression \( 3x \) approaches zero. The fraction \( \frac{1}{3x} \) then rises to very large positive values because the denominator becomes extremely small while remaining positive. This phenomenon where the limit goes to infinity is expressed as \( \lim_{x \to 0^+} \frac{1}{3x} = \infty \).

• Infinite limits help us understand the vertical asymptotic behavior of a function.
• They point to places where the function increases or decreases without restriction as inputs approach a certain point.

Recognizing infinite limits is essential for identifying behaviors in functions that do not settle into a finite value, offering insights into the nature of singularities and growth rates.

Limit Notation

"Limit notation" is the concise language used in calculus to describe the tendency of a function as it approaches a particular input. It formalizes the concept of limits, encapsulating the idea of what value a function gets close to as the input variable heads towards a specified point.
For a limit approaching from the right, we use the notation \( \lim_{x \to a^+} f(x) \), indicating that \( x \) approaches \( a \) from greater values. For instance, \( \lim_{x \to 0^+} \frac{1}{3x} \) specifies that we are taking the limit of the function \( \frac{1}{3x} \) as \( x \) heads towards 0 from numbers larger than 0.

• It provides a precise way to express mathematical ideas involving approximation and boundary behavior.
• This notation is not limited to just real numbers but can extend into more complex areas like multivariable calculus.

Correct usage of limit notation is fundamental in solving calculus problems that involve predicting function behavior at or near undefined points, helping learners grasp the underpinnings of continuous change and conservatively unbounded functions.