Limits and one-sided limits
What a function approaches, even when its value at the point tells a different story.
The idea
Imagine moving along a graph toward a chosen input. Watch the height of the graph as you get closer. If the height settles toward one number, that number is the limit—even if the graph has a hole or a different value at the chosen input.
The key distinction is approaching a point is different from evaluating at the point. A limit may exist even if \(f(a)\) is missing or has a different value.
A practical starting point
For a polynomial, substitute the approaching value. For a quotient of polynomials, this also works when the denominator is nonzero there. If substitution fails, investigate; it is not automatically a failed limit.
Worked examples
Example 1
Direct substitution
Find \(\lim_{x\to 2}(x^2+1)\). A polynomial has no break at the point, so
\[\lim_{x\to 2}(x^2+1)=2^2+1=5.\]Example 2
A hole, not a dead end
Find \(\lim_{x\to 2}\frac{x^2-4}{x-2}\). Direct substitution gives \(0/0\). Instead, factor:
\[\frac{x^2-4}{x-2} =\frac{(x-2)(x+2)}{x-2} =x+2,\qquad x\ne 2.\]Near \(2\), this follows the line in the interactive graph, apart from its value at \(2\). Therefore,
\[\lim_{x\to 2}\frac{x^2-4}{x-2}=\lim_{x\to 2}(x+2)=4.\]Cancellation is valid here because a limit uses nearby points, where \(x-2\ne 0\). It does not define the original quotient at \(2\).
Approaching from one side
Sometimes we restrict where the input comes from. A left-hand limit, written \(\lim_{x\to a^-}f(x)\), uses only inputs smaller than \(a\). A right-hand limit, written \(\lim_{x\to a^+}f(x)\), uses only inputs larger than \(a\).
The ordinary limit we started with is also called a two-sided limit. A finite two-sided limit exists exactly when both one-sided limits exist and agree on the same finite value. Infinite limits describe a different kind of behavior, introduced later in this module.
Example 3
Two sides, two answers
Suppose
\[g(x)=\begin{cases}1,&x<0,\\3,&x\ge 0.\end{cases}\]From the left we stay on the lower branch; from the right we stay on the upper branch. Therefore,
\[\lim_{x\to 0^-}g(x)=1, \qquad \lim_{x\to 0^+}g(x)=3.\]The one-sided limits disagree, so the two-sided limit does not exist. The fact that \(g(0)=3\) cannot repair the jump.
When a limit fails
For a finite limit, the graph must settle toward one height. A disagreement between sides is only one way this can fail: even a single side may never settle.
