Learning Objectives
- Informally define a limit as the value a function approaches.
- Use and interpret limit notation (including one-sided limits).
- Evaluate limits numerically (tables), graphically, and by algebraic simplification.
Why Limits?
Sometimes a function misbehaves at a point—there might be a hole or the value could be undefined. Yet the nearby behavior can still be perfectly predictable. Limits capture that “getting close” idea.
Informal Definition & Notation
We say the limit of \(f(x)\) as \(x\) approaches \(a\) equals \(L\) if the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently close to \(a\) (but not necessarily equal to \(a\)).
Notation: \( \displaystyle \lim_{x\to a} f(x) = L \)
One-sided limits:
- From the left: \( \displaystyle \lim_{x\to a^-} f(x) \)
- From the right: \( \displaystyle \lim_{x\to a^+} f(x) \)
The (two-sided) limit exists iff both one-sided limits exist and are equal.
Example: A Hole Doesn’t Break the Limit
Consider \( \displaystyle f(x)=\frac{x^2-1}{x-1} \). For \( x\ne 1 \), this simplifies to \( f(x)=x+1 \). At \( x=1 \) the original formula is undefined (a “hole”).
- Table approach to \( x=1 \):
| \(x\) | \(f(x)\) | \(x\) | \(f(x)\) |
|---|---|---|---|
| 0.9 | 1.9 | 1.1 | 2.1 |
| 0.99 | 1.99 | 1.01 | 2.01 |
| 0.999 | 1.999 | 1.001 | 2.001 |
Both sides approach \(2\), so \( \displaystyle \lim_{x\to 1} \frac{x^2-1}{x-1} = 2 \).
Interactive Graph: One-Sided Limits & a Hole
Use the slider \(h\) to approach \(a=1\) from the left (\(x=a-h\)) and from the right (\(x=a+h\)). Notice both sides’ \(y\)-values move toward \(2\), even if the point at \(x=1\) is missing or redefined.
The limit at \(x=1\) is \(2\). Changing \(f(1)\) does not change the limit.
Interactive Graph: Jump Discontinuity (Left vs. Right Limits)
Here the function “jumps” at \(a=1\). Adjust the left height \(L\) and right height \(R\). If \(L \ne R\), the two-sided limit does not exist even though both one-sided limits do.
If \( \displaystyle \lim_{x\to 1^-} f(x)=L \) and \( \displaystyle \lim_{x\to 1^+} f(x)=R \) with \(L\ne R\), then the two-sided limit \( \displaystyle \lim_{x\to 1} f(x) \) does not exist.
Limit Laws (Quick Reference)
- Sum/Difference: \( \displaystyle \lim (f \pm g) = \lim f \pm \lim g \)
- Product: \( \displaystyle \lim (fg) = (\lim f)(\lim g) \)
- Quotient: \( \displaystyle \lim \frac{f}{g} = \frac{\lim f}{\lim g} \) (if \( \lim g \ne 0 \))
- If direct substitution gives \( \frac{0}{0} \): factor/cancel or rationalize, then substitute again.
Try It
- Evaluate \( \displaystyle \lim_{x\to 3} \frac{x^2-9}{x-3} \).
- Decide whether \( \displaystyle \lim_{x\to 0^-} \frac{1}{x} \) and \( \displaystyle \lim_{x\to 0^+} \frac{1}{x} \) are equal.
- For the piecewise function \( g(x)=\begin{cases}L,&x<1\\R,&x>1\end{cases} \), determine \( \displaystyle \lim_{x\to 1^-} g(x) \), \( \displaystyle \lim_{x\to 1^+} g(x) \), and explain when \( \displaystyle \lim_{x\to 1} g(x) \) exists.
Key Takeaways
- The limit describes nearby behavior; it may ignore the actual point value.
- Two-sided limit exists iff left and right limits agree.
- A hole is a removable discontinuity; a jump occurs when one-sided limits differ.
- Workflow: substitute → if \( \frac{0}{0} \), simplify → substitute again.