Zoom Out

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Two young mathematicians discuss what curves look like when one ‘‘zooms out.’’

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, think about this function: $f(x) = \frac {1}{x}.$
Riley: Hmmm. If you plot it, the graph looks like this:
Devyn: Right! What I’ve noticed is that if $x$ gets really big or really small, then our function looks like a line.
Riley: Yeah! And if the $x$ values get really close to $0$ , our function also looks like a different line!
Devyn: Whoa! You’re blowin’ my mind.

Devyn and Riley have noticed that the function $f(x) = \frac {1}{x}$ looks like a horizontal line $y \ = \answer {0}$ for $x$ values far away from $0$ and looks like the vertical line $x \ = \answer {0}$ for $x$ values near $0$ .

Let’s investigate why this is the case using some tables.

We’ve noticed that $f(x) = \frac {1}{x}$ looks like a horizontal line for $x$ values far away from $0$ . Fill in the tables below to answer the following question. $\begin {array}{l|l} x & f(x) = \frac {1}{x} \\ \hline 1 & \begin {prompt}\answer {1}\end {prompt}\\ 10 & \begin {prompt}\answer {0.1}\end {prompt}\\ 100 & \begin {prompt}\answer {0.01}\end {prompt}\\ 1000 & \begin {prompt}\answer {0.001}\end {prompt} \\ 10000 & \begin {prompt}\answer {0.0001}\end {prompt} \\ 100000 & \begin {prompt}\answer {0.00001}\end {prompt} \\ \end {array} \qquad \text {and}\qquad \begin {array}{l|l} x & f(x) = \frac {1}{x} \\ \hline -1 & \begin {prompt}\answer {-1}\end {prompt}\\ -10 & \begin {prompt}\answer {-0.1}\end {prompt}\\ -100 & \begin {prompt}\answer {-0.01}\end {prompt}\\ -1000 & \begin {prompt}\answer {-0.001}\end {prompt} \\ -10000 & \begin {prompt}\answer {-0.0001}\end {prompt} \\ -100000 & \begin {prompt}\answer {-0.00001}\end {prompt} \\ \end {array}$

In the first table, as the $x$ values became more positive, $y = \frac {1}{x}$ toward $y = \answer {0}$ . Based on this, it appears that $\displaystyle \lim _{x \to \infty } \frac {1}{x} = \answer {0}$ .
In the second table, as the $x$ values became more negative, $y = \frac {1}{x}$ toward $y = \answer {0}$ . Based on this, it appears that $\displaystyle \lim _{x \to -\infty } \frac {1}{x} = \answer {0}$ .

We’ve also noticed that $f(x) = \frac {1}{x}$ looks like a vertical line for $x$ values near $0$ . Fill in the tables below to answer the following question. $\begin {array}{l|l} x & f(x) = \frac {1}{x} \\ \hline 10 & \begin {prompt}\answer {0.1}\end {prompt}\\ 1 & \begin {prompt}\answer {1}\end {prompt}\\ \frac {1}{10} & \begin {prompt}\answer {10}\end {prompt}\\ \frac {1}{100} & \begin {prompt}\answer {100}\end {prompt} \\ \frac {1}{1000} & \begin {prompt}\answer {1000}\end {prompt} \\ \frac {1}{10000} & \begin {prompt}\answer {10000}\end {prompt} \\ \end {array} \qquad \text {and}\qquad \begin {array}{l|l} x & f(x) = \frac {1}{x} \\ \hline -10 & \begin {prompt}\answer {-0.1}\end {prompt}\\ -1 & \begin {prompt}\answer {-1}\end {prompt}\\ -\frac {1}{10} & \begin {prompt}\answer {-10}\end {prompt}\\ -\frac {1}{100} & \begin {prompt}\answer {-100}\end {prompt} \\ -\frac {1}{1000} & \begin {prompt}\answer {-1000}\end {prompt} \\ -\frac {1}{10000} & \begin {prompt}\answer {-10000}\end {prompt} \\ \end {array}$

In the first table, as the $x$ values got closer to $0$ , $y = \frac {1}{x}$ rapidly . Based on this, it appears that $\displaystyle \lim _{x \to 0^+} \frac {1}{x} = \answer {\infty }$ .
In the second table, as the $x$ values got closer to $0$ , $y = \frac {1}{x}$ rapidly . Based on this, it appears that $\displaystyle \lim _{x \to 0^-} \frac {1}{x} = \answer {-\infty }$ .