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Chapter 2: Problem 28

Find the limits. $$\lim _{x \rightarrow 3^{-}} \frac{x}{x-3}$$

Short Answer

Expert verified
The limit is \(-\infty\).

Step by step solution

01

Understand the Problem

We have the limit \( \lim_{x \to 3^-} \frac{x}{x-3} \) which means we need to evaluate the behavior of the function \( \frac{x}{x-3} \) as \( x \) approaches 3 from the left (indicated by the \( 3^- \) notation).
02

Analyze the Denominator

The denominator \( x - 3 \) becomes \( 0 \) as \( x \) approaches 3, which can cause the function to approach infinity or negative infinity, depending on the direction of approach and the sign of \( x - 3 \).
03

Evaluate the Sign from the Left

When approaching 3 from the left, \( x \) is slightly less than 3, meaning \( x - 3 \) is negative. Thus, as \( x \to 3^- \), the denominator \( x - 3 \to 0^- \).
04

Consider the Behavior of the Function

The numerator \( x \) is positive as \( x \to 3^- \), so \( \frac{x}{x-3} = \frac{+}{-} \) results in the function approaching negative infinity as the denominator goes to zero from the negative side.

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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 refer to the behavior of a function as the variable approaches a specific value from one side only. In our example, we examine the limit as \( x \) approaches 3 from the left, denoted as \( 3^- \). This means we are only concerned about values of \( x \) that are slightly less than 3.
  • When you approach from the left, it鈥檚 similar to watching a car traffic jam where you're only interested in the traffic coming from one side, not the other.
  • This helps us understand how the function behaves when \( x \) gets really close to, but not equal to, a particular value.
One-sided limits are particularly useful in understanding functions that have different behaviors on either side of a point, especially when approaching critical points like asymptotes or discontinuities.
Asymptotic Behavior
Asymptotic behavior in functions is how they behave as they get close to certain lines or points. Specifically, as a variable approaches a point where the function doesn鈥檛 actually reach a finite value, it can head towards infinity or zero. Consider our earlier example:
  • As \( x \to 3^- \), the function \( \frac{x}{x-3} \) doesn鈥檛 settle at a particular number but instead zooms off towards negative infinity.
  • This means there is an asymptote at \( x = 3 \) which is a vertical line where the graph of the function approaches but never actually touches or crosses.
Think of an asymptote as a warning line not to cross. In theory, approaching this line takes the function to extremes but doesn鈥檛 define a specific point on it. Asymptotic behavior is vital for sketching graphs and understanding function properties at points of discontinuity.
Infinite Limits
Infinite limits are observed when a function grows arbitrarily large in magnitude as the input approaches a particular value. When we say the limit is infinite, we mean the function doesn鈥檛 approach a finite limit, rather it heads toward infinity or negative infinity.
  • In our example, as \( x \to 3^- \), the function \( \frac{x}{x-3} \) tends to negative infinity. This is because as \( x \) nears 3 from the left, the denominator \( x-3 \) becomes a tiny negative number, causing the whole fraction to shoot downwards.
  • Infinite limits are indicative of unbounded behavior in specific regions. They help us identify vertical asymptotes or points where the function goes off to infinity.
Understanding infinite limits allows students to predict where a function will tend towards infinity and is crucial for analyzing functions near potential discontinuities or asymptotic lines.

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