Problem 51 Here is the definition of infini... [FREE SOLUTION]

Chapter 2: Problem 51

Here is the definition of infinite right-hand limit. We say that $f(x)$ approaches infinity as $x$ approaches $x_{0}$ from the right, and write $$\lim _{x \rightarrow x_{0}^{+}} f(x)=\infty$$ if, for every positive real number $B$ , there exists a corresponding number $\delta>0$ such that for all $x$ $$x_{0}B$$ Modify the definition to cover the following cases. 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

Expert verified

Modify $\delta$ intervals for left or right approaches; use $B > 0$ for infinity, $B < 0$ for negative infinity.

Step by step solution

Understanding Right-Hand Limit Going to Infinity

The given definition states that the function $ f(x) $ approaches infinity as $ x $ approaches $ x_0 $ from the right. This means for every positive real number $ B $, there exists a $ \delta > 0 $ such that if $ x_0 < x < x_0 + \delta $, then $ f(x) > B $. We are tasked to adapt this definition for other scenarios.

Modifying for Left-Hand Limit Going to Infinity

To say $ \lim_{x \rightarrow x_0^-} f(x) = \infty $, we modify the definition: For every positive real number $ B $, there exists a $ \delta > 0 $ such that for all $ x $, if $ x_0 - \delta < x < x_0 $, then $ f(x) > B $. This means the function approaches infinity as $ x $ gets closer to $ x_0 $ from the left.

Modifying for Right-Hand Limit Going to Negative Infinity

To say $ \lim_{x \rightarrow x_0^+} f(x) = -\infty $, the definition changes to: For every negative real number $ B $ (because we're looking at negative infinity now), there exists a $ \delta > 0 $ such that for all $ x $, if $ x_0 < x < x_0 + \delta $, then $ f(x) < B $. The function goes to negative infinity as $ x $ approaches $ x_0 $ from the right.

Modifying for Left-Hand Limit Going to Negative Infinity

For $ \lim_{x \rightarrow x_0^-} f(x) = -\infty $, the definition is: For every negative real number $ B $, there exists a $ \delta > 0 $ such that for all $ x $, if $ x_0 - \delta < x < x_0 $, then $ f(x) < B $. This means the function approaches negative infinity as $ x $ approaches $ x_0 $ from the left.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right-Hand Limit

The concept of a right-hand limit focuses on how a function behaves as the input approaches a particular value from the right side. Imagine you're walking along the number line towards a point of interest, coming from values that are slightly higher.
That's observing from the 'right-hand' direction. In mathematical terms, we denote this as $ \lim_{x \to x_0^+} f(x) $. The $+$ symbol shows you are approaching from the right.

When we say a function $ f(x) $ approaches infinity, it means that for each vast number $ B $, there is a tiny range to the right of $ x_0 $ where $ f(x) $ exceeds $ B $.

Think of $ \delta $ as a small positive gap where magic happens: specifically, where $ f(x) $ flies above $ B $.
This principle is crucial for understanding behavior as numbers get progressively larger, observing from the right toward the limit point $ x_0 $.

Left-Hand Limit

Left-hand limits are quite similar to right-hand limits, except that you approach from the left side. Imagine moving along a line towards a target number $ x_0 $ but coming from smaller values.
This concept is represented mathematically as $ \lim_{x \to x_0^-} f(x) $, where the negative sign in the superscript indicates approaching from the left.

For a left-hand limit approaching infinity, you want $ f(x) $ to surpass any big number $ B $ as $ x $ closes in on $ x_0 $ from the left (where $ x_0 - \delta < x < x_0 $).

So, $ \delta $ acts as a minuscule step to the left of $ x_0 $, where $ f(x) $ becomes greater than $ B $.
This shows behavior as the function values shoot to infinity as we move from the left of $ x_0 $.

Negative Infinity Limit

When contemplating limits at negative infinity, picture how a function behaves when it dips down towards vast negative values. Both right-hand and left-hand limits can be involved in this scenario.
Mathematically:

Right-hand negative infinity limit is denoted $ \lim_{x \to x_0^+} f(x) = -\infty $.
Left-hand negative infinity limit is denoted $ \lim_{x \to x_0^-} f(x) = -\infty $.

In both of these cases, you look for when $ f(x) $ becomes smaller than any negative $ B $, as defined by the direction it's approaching (from the right or the left).

Here, $ \delta $ remains that crucial small interval that guides the function past the threshold of $ -B $:

For right-hand limits, work with the area to the right of $ x_0 $ ($ x_0 < x < x_0 + \delta $).
For left-hand limits, focus on the space to the left ($ x_0 - \delta < x < x_0 $).

This concept paints a picture of $ f(x) $'s behavior as it explores the depths of negative infinity.

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Short Answer

Step by step solution

Understanding Right-Hand Limit Going to Infinity

Modifying for Left-Hand Limit Going to Infinity

Modifying for Right-Hand Limit Going to Negative Infinity

Modifying for Left-Hand Limit Going to Negative Infinity

Key Concepts

Right-Hand Limit

Left-Hand Limit

Negative Infinity Limit

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