Problem 51 - Calculus and its applications

Question

Is the function given by \(G(x)=\frac{1}{x}\) continuous at \(x=0 ?\) Why or why not?

Step-by-Step Solution

Answer

The function \( G(x) = \frac{1}{x} \) is not continuous at \( x = 0 \) because it is undefined and the limit does not exist.

Step 1: Understand the Continuity Condition

A function \( f(x) \) is continuous at a point \( x = a \) if three conditions are met: 1) \( f(a) \) is defined. 2) The limit of \( f(x) \) as \( x \) approaches \( a \) from both sides exists. 3) The limit of \( f(x) \) as \( x \to a \) is equal to \( f(a) \). Now, evaluate these conditions for \( G(x) = \frac{1}{x} \) at \( x = 0 \).

Step 2: Check If the Function is Defined

The function \( G(x) = \frac{1}{x} \) is not defined at \( x = 0 \) because division by zero is undefined. Therefore, \( G(0) \) does not exist.

Step 3: Evaluate the Limit as x Approaches 0

Determine \( \lim_{x \to 0} \frac{1}{x} \). As \( x \to 0^+ \) (from the right), \( \frac{1}{x} \to +\infty \). As \( x \to 0^- \) (from the left), \( \frac{1}{x} \to -\infty \). Since the one-sided limits are not equal, the overall limit does not exist.

Step 4: Conclusion on Continuity

Since \( G(0) \) is not defined and the limit of \( G(x) \) as \( x \to 0 \) does not exist, \( G(x) = \frac{1}{x} \) is not continuous at \( x = 0 \).

Key Concepts

Limits in Calculus

Limits are a fundamental concept in calculus used to understand the behavior of functions as they approach a specific point or value. They help us explore what a function's output comes close to, even if it's not defined exactly at that point.

• Approaching a Point: When we talk about limits, we refer to how a function behaves as it gets closer and closer to a particular value, let's say \( x = a \).
• Existence of Limits: For a limit to exist at a point, the function should approach the same value from both sides. That means, as \( x \to a^- \) (from the left side) and \( x \to a^+ \) (from the right side), the function should tend towards the same value.

If a function has a defined and equal limit from both sides at a point, that limit governs the function's behavior at the point, even if the function isn't defined there.
In the case of the function \( G(x) = \frac{1}{x} \), as \( x \to 0 \), the limit does not exist because it approaches infinity from different sides, indicating discontinuity.

Undefined Values in Functions

In certain cases, a function may not be defined at specific points due to operations that aren't valid mathematically, such as division by zero. For these cases:

• Identifying Undefined Points: Functions can become undefined when evaluating specific points, like \( x = 0 \) in \( G(x) = \frac{1}{x} \). Here, division by zero occurs, making the expression undefined.
• Effect on Continuity: If a function is undefined at a point, it automatically cannot be continuous there, because the first condition of continuity requires that the function must be defined at the specific point.

These undefined values are crucial in determining whether a function can be continuous at a point. If \( f(a) \) does not exist, the function cannot be described smoothly at that point, reinforcing the discontinuity for \( G(x) = \frac{1}{x} \) at \( x = 0 \).

One-sided Limits

One-sided limits focus on the approach to a specific point from one direction, either from the left or the right. This helps in understanding the behavior of functions, especially near points where the function may not be well-behaved.

• Left-hand Limit: Denoted as \( \lim_{x \to a^-} f(x) \), this represents the value the function approaches as \( x \) comes close to \( a \) from values less than \( a \).
• Right-hand Limit: Denoted as \( \lim_{x \to a^+} f(x) \), this shows the value the function comes to as \( x \) approaches \( a \) from the right (values greater than \( a \)).

For a function's overall limit to exist at a point \( x = a \), these two one-sided limits must be equal. If they differ, like in the function \( G(x) = \frac{1}{x} \) where it approaches \(+\infty\) from the right of 0 and \(-\infty\) from the left, the overall limit does not exist. Thus, without a consistent limit, continuity at that point cannot be established.