Problem 93 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

Modify direction and comparison signs for each case to express left-hand or right-hand limits and infinity directions.

Step 1: Understanding the Original Definition

The original definition states that for any positive number \( B \), there exists a \( \delta > 0 \) such that if \( c < x < c + \delta \), then \( f(x) > B \). This means \( f(x) \) approaches infinity as \( x \) approaches \( c \) from the right.

Step 2: Modify for Case a: Left-hand Limit Approaching Positive Infinity

To modify the definition for \( \lim_{x \to c^{-}} f(x) = \infty \), you need to change the conditions: for every positive real number \( B \), there exists a \( \delta > 0 \) such that if \( c - \delta < x < c \), then \( f(x) > B \). This reflects \( f(x) \) approaching infinity as \( x \) approaches \( c \) from the left.

Step 3: Modify for Case b: Right-hand Limit Approaching Negative Infinity

To cover \( \lim_{x \to c^{+}} f(x) = -\infty \), change the inequality: for every negative real number \( A \), there exists a \( \delta > 0 \) such that if \( c < x < c + \delta \), then \( f(x) < A \). This describes \( f(x) \) approaching negative infinity as \( x \) approaches \( c \) from the right.

Step 4: Modify for Case c: Left-hand Limit Approaching Negative Infinity

For \( \lim_{x \to c^{-}} f(x) = -\infty \), similarly change the conditions: for every negative real number \( A \), there exists a \( \delta > 0 \) such that if \( c - \delta < x < c \), then \( f(x) < A \). This reflects \( f(x) \) approaching negative infinity as \( x \) approaches \( c \) from the left.

Key Concepts

Right-Hand Limit

When discussing limits, the right-hand limit of a function is a crucial concept. It helps in understanding how a function behaves as the variable approaches a specific point from the right side. Imagine you are traveling along a number line towards a particular point, approaching it only from the positive direction. At a mathematical level, the right-hand limit is denoted by \( \lim_{x \to c^+} f(x) \). Here, \( x \) is getting closer to \( c \), but from values that are larger than \( c \).

• To say \( f(x) \to \infty \) as \( x \to c^+ \), is to claim that for any number \( B \), you can find a range such that if \( x \) falls within this range (\( c < x < c + \delta \)), \( f(x) \) is greater than \( B \).

In simpler terms, function values shoot up to infinity as you approach your point from the right side.

Left-Hand Limit

Complementing the right-hand limit, the left-hand limit looks at a function's behavior as it approaches a point from the left. Imagine you're again on that number line, but this time coming towards the point from the left or negative direction. The left-hand limit is represented by \( \lim_{x \to c^-} f(x) \). In this case, \( x \) converges towards \( c \) from values that are smaller than \( c \).

• For \( f(x) \to \infty \) as \( x \to c^- \), it means that for any positive number \( B \), you can choose a range where \( x \) lies within (\( c - \delta < x < c \)), ensuring \( f(x) \) exceeds \( B \).

Here's the simple takeaway: as you come closer to \( c \) from the left, your function values rise indefinitely.

Infinity in Limits

Infinity in limits occurs when the function values increase or decrease without bound as you near a specific point. Such scenarios are pivotal in understanding both vertical asymptotes and unbounded behavior of functions. For instance, if your function moves towards infinity as \( x \) nears \( c \), we write this as \( f(x) \to \infty \). Similarly, negative infinity is depicted by \( f(x) \to -\infty \).

• If \( \lim_{x \to c^+} f(x) = -\infty \), for every negative number \( A \), there's a range such that any \( x \) within it results in \( f(x) \) being less than \( A \).
• For both right-left and negative-infinity scenarios, a precise range \( \delta \) (a gap) allows you to predict if the function dives negatively indefinitely.

Understanding these limits tells you how a function behaves near crucial points, enhancing your grasp of its overall graph characteristics.