Problem 45 - Calculus Early Transcendentals

Question

We write \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\) if for any negative number \(N\) there exists \(\delta>0\) such that $$f(x)

Step-by-Step Solution

Answer

Question: Write the formal definitions for the limit of a function as x approaches a from the right (+) or from the left (-), with the results being positive infinity or negative infinity. Answer: 1. Formal definition for \(\lim _{x \rightarrow a^{+}} f(x)=\infty\): For any positive number N, there exists a positive number \(\delta\) such that if \(0 < x - a < \delta\), then \(f(x) > N\). 2. Formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=-\infty\): For any negative number N, there exists a positive number \(\delta\) such that if \(0 < a - x < \delta\), then \(f(x) < N\). 3. Formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=\infty\): For any positive number N, there exists a positive number \(\delta\) such that if \(0 < a - x < \delta\), then \(f(x) > N\).

Step 1: Understand the given definition for \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\)

The given definition states that for any negative number N, there exists a positive number \(\delta\) such that if \(0 < x - a < \delta\), then \(f(x) < N\). This means when x approaches a from the right (+), the function value decreases without bound.

Step 2: Write an analogous formal definition for \(\lim _{x \rightarrow a^{+}} f(x)=\infty\)

Based on the given definition, we can write the formal definition for \(\lim _{x \rightarrow a^{+}} f(x)=\infty\) as follows: For any positive number N, there exists a positive number \(\delta\) such that if \(0 < x - a < \delta\), then \(f(x) > N\). b. Write an analogous formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=-\infty\)

Step 1: Identify the change from the right limit to the left limit

The main difference when changing from a right limit to a left limit is that we are approaching a from the left side (-) rather than the right side (+). This means the inequality becomes \(0 < a - x < \delta\).

Step 2: Write an analogous formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=-\infty\)

Based on the given definition and the change identified in step 1, we can write the formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=-\infty\) as follows: For any negative number N, there exists a positive number \(\delta\) such that if \(0 < a - x < \delta\), then \(f(x) < N\). c. Write an analogous formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=\infty\)

Step 1: Combine the changes for the left limit and positive infinity

In this case, we need to combine the changes from the previous two parts. We are approaching a from the left side (-) and the function value is approaching positive infinity.

Step 2: Write an analogous formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=\infty\)

Based on the given definition and the changes identified in the previous steps, we can write the formal definition for \(\lim _{x \rightarrow a^{-}} f(x)=\infty\) as follows: For any positive number N, there exists a positive number \(\delta\) such that if \(0 < a - x < \delta\), then \(f(x) > N\).

Key Concepts

Right-Hand Limit

Understanding the right-hand limit means focusing on how a function behaves as the input approaches a particular point from the right. When we say \( \lim_{x \to a^+} f(x) = L \), it means that as the x-values get closer and closer to \(a\) from values greater than \(a\), the function value \(f(x)\) approaches \(L\).

• "\(x \to a^+\)" specifically indicates that x is approaching a from the right-hand side.
• The fact that \(f(x) = L\) signifies that the function settles or almost gets near to this value \(L\).

This concept is crucial when examining one-sided limits and plays a significant role in determining how functions behave at points where they're not well-defined. It is especially useful when dealing with piecewise functions where behavior might differ depending on the side from which \(x\) approaches \(a\).

Left-Hand Limit

The left-hand limit is similar but now considers the behavior of the function as \(x\) approaches from the left-hand side, or from values less than \(a\). Mathematically, this is expressed as \(\lim_{x \to a^-} f(x) = L\).

• For \(x \to a^-\), the focus is on x-values that are coming closer from left of \(a\).
• Similarly, \(f(x) = L\) denotes that the function approaches a specific real number \(L\).

Left-hand limits are significant when studying limits at discontinuities or corners in graphs.In practice, left-hand and right-hand limits must be equal for the overall limit to exist at \(a\). This concept represents how crucial understanding function behavior from both sides can be in analyzing functions comprehensively.

Infinite Limits

Infinite limits describe scenarios where a function doesn't settle to a real finite number as the variable approaches a given point but instead increases or decreases without bound. When we express \( \lim_{x \to a} f(x) = \pm \infty \), it essentially highlights that as \(x\) approaches \(a\), the function variable grows positively or negatively beyond any finite value:

• If \(f(x) > N\) for positive infinity, where \(N\) is any large positive number.
• Conversely, \(f(x) < N\) for any large negative \(N\) suggests negative infinity.

These concepts are key when examining asymptotic behavior, particularly in rational functions and vertical asymptotes. Recognizing infinite limits is crucial as they denote that parts of a graph extend towards infinity, forming key characteristics like vertical asymptotes.

Delta-Epsilon Definitions

The delta-epsilon definition is a foundational idea in calculus to rigorously describe limits. This involves specifying how close \(x\) needs to be to \(a\) (using delta \(\delta\)) to force \(f(x)\) within a specific distance to the limit \(L\) (using epsilon \(\epsilon\)). In essence, for \( \lim_{x \to a} f(x) = L \):

• "For every \(\epsilon > 0\) (no matter how small the range around \(L\)), there exists a \(\delta > 0\)."
• This means that if \(0 < |x-a| < \delta\), then \(|f(x) - L| < \epsilon\).

This definition underpins the formal, rigorous approach to limits, not just suggesting but proving how limits work in mathematical terms. It's a powerful mathematical tool that ensures functions behave as expected using observable metrics, providing clarity on what it means when a function nears a specific value.