Question

Define (a) \(\lim _{x \rightarrow x_{0}-} f(x)=-\infty\) (b) \(\lim _{x \rightarrow x_{0}+} f(x)=\infty\) (c) \(\lim _{x \rightarrow x_{0}+} f(x)=-\infty\)

Short answer

1. Define the limit expression (a) \(\lim _{x \rightarrow x_{0}-} f(x)=-\infty\). The limit expression (a) represents a function that approaches negative infinity as x approaches a specific point, \(x_0\), from the left. This signifies that as x gets slightly smaller than \(x_0\), the value of f(x) becomes increasingly large in magnitude and negatively oriented.

Step-by-step solution

(a) Definition for \(\lim _{x \rightarrow x_{0}-} f(x)=-\infty\)#

The limit expression \(\lim _{x \rightarrow x_{0}-} f(x)=-\infty\) signifies that the function \(f(x)\) approaches negative infinity as \(x\) approaches \(x_0\) from the left. In other words, when \(x\) is slightly smaller than \(x_0\), \(f(x)\) becomes increasingly large in magnitude and negatively oriented.

(b) Definition for \(\lim _{x \rightarrow x_{0}+} f(x)=\infty\)#

The limit expression \(\lim _{x \rightarrow x_{0}+} f(x)=\infty\) indicates that the function \(f(x)\) approaches positive infinity as \(x\) approaches \(x_0\) from the right. This means that as \(x\) gets very slightly larger than \(x_0\), \(f(x)\) increases without bound, going towards positive infinity.

(c) Definition for \(\lim _{x \rightarrow x_{0}+} f(x)=-\infty\)#

The limit expression \(\lim _{x \rightarrow x_{0}+} f(x)=-\infty\) represents that the function \(f(x)\) approaches negative infinity as \(x\) approaches \(x_0\) from the right. In simpler terms, when \(x\) is just marginally greater than \(x_0\), \(f(x)\) decreases without bound towards negative infinity.

Key concepts

Left-Hand Limit

The left-hand limit refers to the behavior of a function as the input, denoted by \(x\), nears a certain value, \(x_0\), from the left side. This means that we are observing the changes in the function as \(x\) takes values just slightly less than \(x_0\). In math terms, the left-hand limit is written as \( \lim_{x \to x_0^-} f(x) \).

For example, given the expression \( \lim _{x \rightarrow x_{0}-} f(x)=-\infty \), it shows that as \(x\) approaches \(x_0\) from values less than \(x_0\), the function \(f(x)\) will dive towards negative infinity. It's like saying if you keep a close watch approaching a cliff (\(x_0\)), you're stepping back as the land drops drastically downward (towards \(-\infty\)).

In simpler terms, if you have a curve and you are watching it from the left heading towards a point \(x_0\), a left-hand limit helps you predict if the curve shoots down indefinitely as you get closer.

Right-Hand Limit

The right-hand limit focuses on understanding how a function behaves when the input variable \(x\) gets very close to a particular point \(x_0\) but from the right. Essentially, this means \(x\) is just a bit larger than \(x_0\). We denote this with the symbol \(\lim_{x \to x_0^+} f(x)\).

In the expression \(\lim_{x \to x_0^+} f(x) = \infty\), it indicates that as \(x\) is slightly greater than \(x_0\), \(f(x)\) grows without limit in a positive direction. This is similar to an elevator rising as it moves past a certain floor, going upwards and upwards without a stopping point.

Similarly, if the expression is \(\lim_{x \to x_0^+} f(x) = -\infty\), it would mean that the function \(f(x)\) plummets negatively, falling steeply downward as \(x\) closes in on \(x_0\) from larger values, like descending into a deep canyon.

Infinite Limits

Infinite limits are intriguing aspects of a function's behavior where the function increases or decreases without any boundary as the input approaches some value. Infinite limits can be either positive or negative, based on whether \(f(x)\) is shooting upwards towards \(\infty\) or spiraling down towards \(-\infty\).

- **Positive Infinite Limit:** When the limit of \(f(x)\) as \(x\) approaches \(x_0\) is \(\infty\), the function grows indefinitely in the positive direction. It's like a kite in a gentle wind, rising higher and higher as time goes on.- **Negative Infinite Limit:** Conversely, if the limit is \(-\infty\), the function plunges ever downward. Imagine a stone falling into a bottomless pit, forever descending.

Infinite limits help in understanding points of discontinuity or vertical asymptotes in functions. They're especially useful in evaluating functions near undefined points, describing the wild ride a function might take as it veers far from typical behavior.