\[ f(x) = \frac {x}{x}. \]
\[ f(0) = \answer {DNE}\qquad \lim _{x\to 0} f(x) = \answer {1}. \]
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
Consider the function
\[ f(x) = \frac {4x}{x}. \]
\[ f(0) = \answer {DNE}\qquad \lim _{x\to 0} f(x) = \answer {4}. \]
Consider the function
2025-01-06 19:41:46 \[ f(x) = \frac {x}{-3x}. \]
\[ f(0) = \answer {DNE}\qquad \lim _{x\to 0} f(x) = \answer {-1/3}. \]