Problem 28 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If the graph of \(f\) has a vertical asymptote at \(x=1,\) then \(f\) cannot be continuous at \(x=1\)

Step-by-Step Solution

Answer

The statement is true because a vertical asymptote means the function is discontinuous at that point.

Step 1: Understanding Vertical Asymptotes

A vertical asymptote at \(x = 1\) for a function \(f\) means that as \(x\) approaches 1 from either left or right, the function value \(f(x)\) tends towards infinity or negative infinity. This suggests a break or discontinuity in the function at that point.

Step 2: Evaluating Continuity

For a function \(f\) to be continuous at a point \(x = c\), three conditions must be satisfied: \(f(c)\) is defined, the limit as \(x\) approaches \(c\) from both sides exists, and the limit equals \(f(c)\).

Step 3: Checking the Continuity Conditions at x=1

Given that \(f\) has a vertical asymptote at \(x = 1\), it implies that the limit \(\lim_{x \to 1} f(x)\) does not exist (it goes to \(\pm \infty\)). Therefore, one of the conditions for continuity fails, specifically that the limit must equal \(f(1)\), making the function discontinuous at \(x=1\).

Step 4: Conclusion

Since the required conditions of continuity cannot be met at \(x=1\), the statement "If the graph of \(f\) has a vertical asymptote at \(x=1,\) then \(f\) cannot be continuous at \(x=1\)" is true.

Key Concepts

Vertical Asymptotes

When you hear about vertical asymptotes, think of them as invisible lines that a function's graph approaches but never touches. A vertical asymptote arises in the graph of a function when, as you move closer and closer to a certain x-value, the function values zoom up to infinity or plunges down to negative infinity.
Vertical asymptotes occur because something in the function makes it impossible to give a finite output near certain points. Often, they are caused by division by zero which doesn't resolve to a nice number.
Asymptotes communicate important information about the behavior of a function. They tell us that the function does not exist in a meaningful, finite sense at some points, making the function exhibit a kind of wild, extreme behavior.

Function Discontinuity

Functions are continuous if you can trace them from one side of a point to the other without lifting your pencil. But what happens if there's a sudden jump or "hole" in the graph? That's where function discontinuities come into play.
Discontinuity means the function isn't smooth at certain points. Instead of flowing neatly, the graph jumps or breaks off. This can be due to reasons like vertical asymptotes or point removals where functions are undefined.
Key types of discontinuities include:

• Jump Discontinuity: Sudden jumps from one line to another.
• Infinite Discontinuity: Characterized by vertical asymptotes where the function heads to infinity.
• Removable Discontinuity: Functions that have a "hole," usually due to factor cancellations.

Understanding when and why a function is discontinuous helps us predict certain behaviors, like where a graph might explode to infinity, or simply not exist.

Limits in Calculus

Calculus is like peeking very closely at what's happening with functions at certain "borderline" points. Limits are our handy tools for understanding behaviors around tricky spots. Imagine squeezing in closer from the left and right of some point. The value you approach is the limit.
Limits help us explore and solve problems about continuity, but they also apply to differentiation and integration. Calculating limits allows us to precisely define:**

• **Continuity:** Is the function smooth, or are there interruptions?
• **Derivatives:** What's the function's rate of change at a point?
• **Integrals:** What's the total change over an interval?

In the context of vertical asymptotes, limits confirm that as you near the asymptote, the function values bolt to infinity, explaining why true continuity fails at those points. Using limits, we gain deeper insights that unravel potential mysteries behind the graphed squiggles and curves.