Problem 55 - Calculus: Graphical, Numerical, Algebraic

Question

Group Activity In Exercises 55 and \(56,\) sketch a graph of a function \(y=f(x)\) that satisfies the stated conditions. Include any asymptotes. $$\begin{array}{l}{\lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 5^{-}} f(x)=\infty, \lim _{x \rightarrow 5^{+}} f(x)=\infty} \\ {\lim _{x \rightarrow \infty} f(x)=-1, \quad \lim _{x \rightarrow-2^{+}} f(x)=-\infty} \\\ {\lim _{x \rightarrow-2^{-}} f(x)=\infty, \lim _{x \rightarrow-\infty} f(x)=0}\end{array}$$

Step-by-Step Solution

Answer

The function starts from the top of the graph for \(x < -2\) and drops down towards negative infinity as it approaches \(x = -2\) from the right. Between \(x = -2\) and \(x = 1\), the function passes through the point (1,2) before increasing sharply to positive infinity at \(x = 5\). As \(x\) goes towards positive infinity, the function gradually converges to \(y = -1\). This behavior defines the graph of the function.

Step 1: Understanding the limits and asymptotes

Firstly, it's important to understand what each limit means for the graph:\nThe limit as \(x\) approaches 1 is 2, hence the function approaches the value of 2 when it gets near '1'.\nThe limit as \(x\) approaches 5 from the left (5-) and from the right (5+) is infinity, so there's likely a vertical asymptote at \(x = 5\) since \(f(x)\) tends to infinity there.\nThe limit as \(x\) approaches infinity is -1, hence as \(x\) gets larger and larger, the function is getting closer and closer to -1.\nThe limit as \(x\) approaches -2 from the right (-2+) is negative infinity and from the left (-2-) is infinity, presenting another vertical asymptote at \(x = -2\).\nFinally the limit as \(x\) approaches negative infinity is 0, showing the function gradually approaches 0 as \(x\) becomes more and more negative.

Step 2: Drawing the asymptotes

Vertical asymptotes occur at \(x = 5\) and \(x = -2\) and there should be a horizontal asymptote at \(y = -1\) since \(f(x)\) approaches -1 as \(x\) approaches infinity and another at \(y = 0\) as \(f(x)\) approaches 0 when \(x\) goes to negative infinity.

Step 3: Sketching the function

After setting up asymptotes, now the sketching can be started. Next to the left of \(x = -2\), \(f(x)\) approaches infinity, hence the function should start from the top of the graph. The function then goes to negative infinity as it approaches -2 from the right. Between \(x = -2\) and \(x = 1\), the function has to pass through \(y = 2\) at \(x = 1\). After which, it should move towards positive infinity at \(x = 5\) from either direction. Finally, as \(x\) goes to positive infinity, the function should converge to \(y = -1\).

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. These lines can either be vertical, horizontal, or even slant, depending on the behavior of the function. In the context of the problem, understanding asymptotes is crucial since they help outline the movement and boundaries of the function.

Here's a quick rundown of the key types:

• Vertical Asymptotes: These occur at values of \(x\) where the function tends towards infinity. If \(f(x)\) spikes upwards or downwards near a particular \(x\) value, look for a vertical asymptote there.
• Horizontal Asymptotes: These represent a value that \(f(x)\) approaches as \(x\) becomes very large or very small. They provide an idea of the end behavior of the function.

In this exercise, the function has vertical asymptotes at \(x = 5\) and \(x = -2\), whereas horizontal asymptotes are found when \(y = -1\) and \(y = 0\). Recognizing these helps in predicting how the graph behaves near these key points.

Graph Sketching

Graph sketching is about visualizing a function based on its behavior. By using information such as limits, asymptotes, and key points, we can draw a general path of the graph.

Here's what you need to consider:

• Identify asymptotes, as they will guide the boundaries of the graph.
• Determine significant points through limit evaluation, like where the curve meets or crosses usual points.
• Combine the behavior from these elements to sketch a curve that matches the function's characteristics.

In the provided solution, the sketch starts with understanding how the function navigates around asymptotes and particular points, like \(y=2\) when \(x=1\), eventually conforming to the asymptotic boundaries as \(x\) moves through the specified regions.

Infinite Limits

Infinite limits describe the behavior of functions as they approach certain points, where the values either shoot up to infinity or plummet to negative infinity. This is especially important in identifying vertical asymptotes.

For instance:

• When \( \lim_{x \rightarrow 5^-} f(x) = \infty \) and \( \lim_{x \rightarrow 5^+} f(x) = \infty \), the implication is that the function grows immensely in either direction as it nears \(x = 5\).
• Similarly, \( \lim_{x \rightarrow -2^+} f(x) = -\infty \) and \( \lim_{x \rightarrow -2^-} f(x) = \infty \) suggests drastic changes in direction around \(x = -2\).

These infinite limits indicate vertical asymptotes and highlight where the function might exhibit extreme growth or decay.

Horizontal Asymptotes

Horizontal asymptotes indicate a line that the curve tends to as \(x\) moves to infinity either positively or negatively. They are crucial for understanding the long-term trend of a function, essentially describing what happens to \(f(x)\) far away from the vertical limits.

In simpler terms:

• \( \lim_{x \rightarrow \infty} f(x) = -1 \) indicates that as \(x\) becomes very large, the function levels off towards \(y = -1\).
• On the flip side, the function also approaches \(y = 0\) as \(x\) moves towards negative infinity, which is seen through \( \lim_{x \rightarrow -\infty} f(x) = 0 \).

These limits mean that the function has horizontal asymptotes at both \(y = -1\) and \(0\), setting expectations for the graph's end behavior.