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. $$\begin{array}{l}{\lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 5^{-}} f(x)=\infty, \lim _{x \rightarrow 5^{+}} f(x)=\infty} \\ {\lim _{x \rightarrow \infty} f(x)=-1, \quad \lim _{x \rightarrow-2^{+}} f(x)=-\infty} \\\ {\lim _{x \rightarrow-2^{-}} f(x)=\infty, \lim _{x \rightarrow-\infty} f(x)=0}\end{array}$$

Short 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-by-step solution

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.

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.

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 \(y = -1\).