CHANGE AND LIMITS 01
Limits
A limit describes the value a function approaches as its input approaches a particular value.
The main idea
The statement
\[\boxed{\lim_{x\to a}f(x)=L}\]means that \(f(x)\) approaches \(L\) as \(x\) approaches \(a\). The behaviour near \(a\) matters; the value exactly at \(a\) may be a separate question.
1. Approaching a value
For \(f(x)=x^2\), as \(x\to2\), the outputs approach \(4\):
\[1.9^2=3.61,\quad1.99^2=3.9601,\quad2.01^2=4.0401.\]Thus
\[\boxed{\lim_{x\to2}x^2=4}.\]2. Reading the notation
In \(\lim_{x\to a}f(x)=L\), \(x\to a\) describes the input approaching \(a\), while \(f(x)\to L\) describes the output approaching \(L\).
Read it: “the limit of \(f(x)\) as \(x\) tends to \(a\) is \(L\).”
\(x\to a\)
↓ through \(f\)
output
\(f(x)\to L\)
3. Numerical and graphical evidence
Values from both sides can be used to estimate a limit:
\[\begin{array}{c|ccccc}x&1.9&1.99&1.999&2.001&2.01\\x^2&3.61&3.9601&3.996001&4.004001&4.0401\end{array}\]The graph provides the same information geometrically: points on the curve approach \((2,4)\) from both sides.
4. A limit can exist at a missing point
Consider
\[f(x)=\frac{x^2-4}{x-2},\qquad x\ne2.\]For \(x\ne2\), \(f(x)=x+2\). The original expression is undefined at \(2\), but nearby values approach \(4\):
\[\boxed{\lim_{x\to2}\frac{x^2-4}{x-2}=4}.\]5. Function value versus limit
\(f(a)\) asks what happens exactly at \(a\). The limit asks what happens nearby.
A function can therefore have \(f(a)\ne L\) while
\[\lim_{x\to a}f(x)=L.\]The graph illustrates a curve approaching one height while a filled point gives the function a different value.
6. Left- and right-hand limits
Approaching from smaller inputs is
\[\lim_{x\to a^-}f(x),\]and approaching from larger inputs is
\[\lim_{x\to a^+}f(x).\]A two-sided limit exists only if these approach the same value.
7. When a limit does not exist
For
\[f(x)=\begin{cases}1,&x<0,\\3,&x\ge0,\end{cases}\]the one-sided limits are
\[\lim_{x\to0^-}f(x)=1,\qquad\lim_{x\to0^+}f(x)=3.\]They disagree, so the two-sided limit does not exist.
8. Finding limits algebraically
Direct substitution works when it gives an ordinary value:
\[\lim_{x\to3}(x^2+2x)=15.\]If substitution produces \(0/0\), simplify first. For example,
\[\frac{x^2-9}{x-3}=x+3\quad(x\ne3),\]hence
\[\boxed{\lim_{x\to3}\frac{x^2-9}{x-3}=6}.\]↓
ordinary value → evaluate
\(0/0\) → simplify first
↓
then evaluate the limit
9. Infinite limits
For
\[f(x)=\frac1{x^2},\]the function grows without bound as \(x\to0\):
\[\boxed{\lim_{x\to0}\frac1{x^2}=+\infty}.\]The line \(x=0\) is a vertical asymptote.
10. Limits at infinity
Limits also describe long-run behaviour. For
\[f(x)=\frac{2x+1}{x+1},\]divide numerator and denominator by \(x\):
\[\frac{2+1/x}{1+1/x}\to2.\]Therefore
\[\boxed{\lim_{x\to\infty}\frac{2x+1}{x+1}=2}.\]11. Limits and rates of change
For two points on \(y=f(x)\),
\[\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}\]is the secant slope. As \(\Delta x\to0\), the second point approaches the first and the secant approaches the tangent.
12. From limits to derivatives
The derivative is defined by
\[\boxed{f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}}.\]For \(f(x)=x^2\), the quotient simplifies to \(2x+h\), so as \(h\to0\),
\[\boxed{f'(x)=2x}.\]A limit describes nearby behaviour and can be studied numerically, graphically and algebraically. It may exist even when the function is undefined at the point. One-sided limits determine two-sided limits, limits describe unbounded and long-run behaviour, and the limiting secant slope leads directly to the derivative.