Question

draw the graph of a function with both of the following properties. type anything in the box and show your graph to the camera.

• $lim_{x

ightarrowinfty}f(x)=1$

• $lim_{x

ightarrow0^{+}}f(x)=+infty$

Explanation:

Step1: Recall limit - behavior concepts

The limit $\lim_{x
ightarrow\infty}f(x) = 1$ means the function has a horizontal asymptote at $y = 1$ as $x$ goes to positive or negative infinity. The limit $\lim_{x
ightarrow0^{+}}f(x)=+\infty$ means the function has a vertical asymptote at $x = 0$ and the function values increase without bound as $x$ approaches $0$ from the right - hand side.

Step2: Choose a function form

A rational function of the form $f(x)=\frac{1}{x}+1$ satisfies the given conditions.
As $x
ightarrow\infty$, $\lim_{x
ightarrow\infty}(\frac{1}{x}+1)=\lim_{x
ightarrow\infty}\frac{1}{x}+\lim_{x
ightarrow\infty}1 = 0 + 1=1$.
As $x
ightarrow0^{+}$, $\lim_{x
ightarrow0^{+}}(\frac{1}{x}+1)=\lim_{x
ightarrow0^{+}}\frac{1}{x}+\lim_{x
ightarrow0^{+}}1=+\infty + 1=+\infty$.

Step3: Sketch the graph

1. Draw the horizontal asymptote $y = 1$ as a dashed line.
2. Draw the vertical asymptote $x = 0$ as a dashed line.
3. For $x>0$, as $x$ gets larger, the function $y=\frac{1}{x}+1$ approaches the horizontal asymptote $y = 1$ from above. As $x$ approaches $0$ from the right, the function values increase without bound.

Answer:

A function like $y=\frac{1}{x}+1$ can be graphed to satisfy the given limit conditions. The graph has a horizontal asymptote at $y = 1$ and a vertical asymptote at $x = 0$, and the function approaches $y = 1$ as $x
ightarrow\infty$ and approaches $+\infty$ as $x
ightarrow0^{+}$.