Problem 53 - Thomas Calculus

Question

Find the limits in Exercises \(53-58.\) $$\lim \frac{1}{x^{2}-4}\text{ as }$$ \begin{equation}\begin{array}{ll}{\text { a. }} & {x \rightarrow 2^{+}} & {\text { b. } x \rightarrow 2^{-}} \\ {\text { c. }} & {x \rightarrow-2^{+}} & {\text { d. } x \rightarrow-2^{-}}\end{array}\end{equation}

Step-by-Step Solution

Answer

(a) +∞, (b) -∞, (c) -∞, (d) +∞.

Step 1: Identify the Function

The function given is \( f(x) = \frac{1}{x^2 - 4} \). Our goal is to determine the limit of \( f(x) \) as \( x \) approaches specific values. This will require us to consider the behavior of the denominator \( x^2 - 4 \) as \( x \) approaches each value.

Step 2: Analyze Asymptotic Behavior for x Approaching from the Right (Positive)

Let's start with \( a. \) when \( x \rightarrow 2^{+} \). As \( x \) approaches 2 from the right, \( x = 2^+ \), the expression \( x^2 - 4 \) becomes a very small positive number because \( x^2 \) is slightly more than 4. Hence, \( \frac{1}{x^2 - 4} \) tends towards positive infinity. Thus, \[ \lim_{{x \to 2^{+}}} \frac{1}{x^2 - 4} = +\infty \]

Step 3: Analyze Asymptotic Behavior for x Approaching from the Left (Negative)

Now for \( b. \), when \( x \rightarrow 2^{-} \). As \( x \) approaches 2 from the left, \( x = 2^- \), \( x^2 - 4 \) becomes a very small negative number, since \( x^2 \) is slightly less than 4. Hence, \( \frac{1}{x^2 - 4} \) tends towards negative infinity. \[ \lim_{{x \to 2^{-}}} \frac{1}{x^2 - 4} = -\infty \]

Step 4: Analyze Asymptotic Behavior for x Approaching from the Right at Negative Value

Consider \( c. \) when \( x \rightarrow -2^{+} \). As \( x \) approaches -2 from the right, \( x = -2^+ \), we find that \( x^2 - 4 \) becomes a small negative number because \( x^2 \) is close to 4 but less than it. Therefore, \( \frac{1}{x^2 - 4} \) will tend towards negative infinity. \[ \lim_{{x \to -2^{+}}} \frac{1}{x^2 - 4} = -\infty \]

Step 5: Analyze Asymptotic Behavior for x Approaching from the Left at Negative Value

Finally, for \( d. \) where \( x \rightarrow -2^{-} \). As \( x \) approaches -2 from the left, \( x = -2^- \), \( x^2 - 4 \) becomes a small positive number since \( x^2 \) is slightly more than 4. Consequently, \( \frac{1}{x^2 - 4} \) tends to positive infinity. \[ \lim_{{x \to -2^{-}}} \frac{1}{x^2 - 4} = +\infty \]

Key Concepts

Limits

In calculus, the limit of a function describes the behavior of the function's output as the input approaches a certain value. Limits are vital for understanding how functions behave near specific points, even if they do not reach those points. For example, when approaching 2 from either side in the function \( f(x) = \frac{1}{x^2 - 4} \), the output behavior becomes noticeable. If we consider the function as \( x \to 2^{+} \), the denominator becomes a very tiny positive value, making the fraction approach positive infinity. On the other hand, as \( x \to 2^{-} \), the denominator is a tiny negative value, which makes the fraction approach negative infinity.

• Limits help identify whether a function approaches a specific finite value or diverges to infinity.
• They indicate if the function possesses continuity near the point of interest.
• The behavior of the limit varies based on the direction of approach (right or left).

Understanding limits provides a foundation for more advanced calculus concepts like continuity, derivatives, and integrals.

Asymptotic Behavior

Asymptotic behavior explains how a function behaves as it approaches a point or extends to infinity. In the context of the function \( f(x) = \frac{1}{x^2 - 4} \), it is crucial to examine as \( x \) gets close to values like 2 or -2. This function has vertical asymptotes at \( x = 2 \) and \( x = -2 \), where the denominator equals zero, causing the function to diverge to either positive or negative infinity.

• Vertical asymptotes indicate that the function's value grows indefinitely as it approaches a given \( x \) value.
• The behavior is determined by examining the numerator and the sign of the denominator as \( x \) approaches the asymptote.
• Understanding asymptotic behavior is essential in graphing functions and evaluating infinite limits.

Recognizing asymptotes helps anticipate the growth or decay rate near these critical points, depicting the overall shape of the graph and its behavior at extreme points.

One-Sided Limits

One-sided limits focus on the behavior of a function as it approaches a specific point from one direction: the left or the right. It allows a finer analysis when a function behaves differently on either side of a point. For instance, in \( f(x) = \frac{1}{x^2 - 4} \) as \( x \) approaches 2, the limit from the right, \( x \rightarrow 2^{+} \), diverges to positive infinity, while the limit from the left, \( x \rightarrow 2^{-} \), diverges to negative infinity.

• One-sided limits are noted with the superscripts \( + \) (from right) or \( - \) (from left).
• These limits provide insight into the slope and directionality of a graph at specific points.
• They are crucial for determining the presence of a jump or other discontinuity in a graph.

Mastering one-sided limits enhances the understanding of discontinuities and conjunctive points in calculus, thereby building toward deeper mathematical concepts such as derivatives and rate of change.