Could it be anything?

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Two young mathematicians investigate the arithmetic of large and small numbers.

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

Devyn: Hey Riley, remember $\lim _{\theta \to 0}\frac {\sin (\theta )}{\theta }?$
Riley: It is equal to $1$ !
Devyn: But was that crazy proof with all the triangles really necessary? I mean, just plug in zero. $\bigg [\frac {\sin (\theta )}{\theta }\bigg ]_{\theta =0} = \frac {\sin (0)}{0}=\frac {0}{0}\dots$
Riley: You were going to say “ $1$ ,” right?
Devyn: Yeah, but now I’m not sure I was right.
Riley: Dividing by zero is usually a bad idea.
Devyn: You are right. I will never do it again! Also, don’t tell anyone about this conversation.
Riley: What conversation?
Devyn: Exactly.

Consider the function $f(x) = \frac {x}{x}.$ $f(0) = \answer {DNE}\qquad \lim _{x\to 0} f(x) = \answer {1}.$

Consider the function $f(x) = \frac {4x}{x}.$ $f(0) = \answer {DNE}\qquad \lim _{x\to 0} f(x) = \answer {4}.$

Consider the function $f(x) = \frac {x}{-3x}.$ $f(0) = \answer {DNE}\qquad \lim _{x\to 0} f(x) = \answer {-1/3}.$