CHANGE AND LIMITS 02
One-Sided Limits
A one-sided limit asks what a function approaches when the input approaches a point from only one direction.
The main idea
Near the same input \(a\), a function can behave differently on its left and right sides. We therefore distinguish
\[\boxed{\lim_{x\to a^-}f(x)}\qquad\text{and}\qquad\boxed{\lim_{x\to a^+}f(x)}.\]The superscript tells us the direction of approach, not the sign of \(a\).
1. Approaching from the left and right
To approach \(2\) from the left, use values smaller than \(2\):
\[1.9,\ 1.99,\ 1.999,\ldots\to2.\]To approach from the right, use values larger than \(2\):
\[2.1,\ 2.01,\ 2.001,\ldots\to2.\]In both cases the values get arbitrarily close to \(2\), but from different directions.
2. Reading the notation correctly
The statement
\[\lim_{x\to a^-}f(x)=L\]means that \(f(x)\to L\) while \(x\) approaches \(a\) through values \(x<a\).
Similarly,
\[\lim_{x\to a^+}f(x)=L\]uses values \(x>a\). Thus \(a^-\) means “from the left” and \(a^+\) means “from the right”.
→ \(x\to a^-\)
target \(a\)
\(x>a\)
→ \(x\to a^+\)
3. Example where both sides agree
For \(f(x)=x^2\), approaching \(2\) from either side gives outputs approaching \(4\):
\[\lim_{x\to2^-}x^2=4,\qquad\lim_{x\to2^+}x^2=4.\]The graph approaches the same height from both directions.
4. When does the ordinary limit exist?
The two-sided limit
\[\lim_{x\to a}f(x)=L\]exists precisely when the left- and right-hand limits both exist and have the same value:
\[\boxed{\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L}.\]This is why checking both sides is essential when a graph changes behaviour near a point.
+
right-hand limit \(=L\)
↓
two-sided limit \(=L\)
5. A jump: the two sides disagree
Consider
\[f(x)=\begin{cases}1,&x<0,\\3,&x\ge0.\end{cases}\]Then
\[\lim_{x\to0^-}f(x)=1,\qquad\lim_{x\to0^+}f(x)=3.\]Since \(1\ne3\), there is no single value approached from both sides:
\[\boxed{\lim_{x\to0}f(x)\text{ does not exist}.}\]6. The function value does not decide the limit
In the jump example, \(f(0)=3\) because the filled point belongs to the right-hand branch. But this does not change
\[\lim_{x\to0^-}f(x)=1.\]A limit describes values near the point. The actual value \(f(a)\) is a separate question.
7. A one-sided limit at a domain boundary
For the real function
\[f(x)=\sqrt{x},\qquad x\ge0,\]there are no domain values immediately to the left of \(0\). The meaningful approach is therefore from the right:
\[\boxed{\lim_{x\to0^+}\sqrt{x}=0}.\]This is an important reason one-sided limits are needed: sometimes only one side of a point belongs to the domain.
8. A piecewise function that still has a limit
A change of formula does not automatically destroy a limit. Let
\[f(x)=\begin{cases}x+1,&x<1,\\2x,&x\ge1.\end{cases}\]From the left, \(x+1\to2\). From the right, \(2x\to2\). Hence
\[\boxed{\lim_{x\to1}f(x)=2}.\]The formulas differ, but their values meet at the same limiting height.
9. How to read one-sided limits from a graph
To find \(\lim_{x\to a^-}f(x)\), trace the graph toward \(x=a\) from the left and watch the height it approaches. For \(\lim_{x\to a^+}f(x)\), do the same from the right.
Do not begin by looking at the filled point at \(x=a\). First ask what each branch approaches.
compare the two heights
trace right branch → limiting height
same → two-sided limit exists
different → it does not
10. Why one-sided limits matter
One-sided limits let us distinguish behaviour that an ordinary limit can hide. They are especially useful for piecewise functions, jumps, endpoints and vertical asymptotes.
They also give the basic test for a two-sided limit:
\[\boxed{\text{left limit}=\text{right limit}\quad\Longleftrightarrow\quad\text{two-sided limit exists}}\]provided the two one-sided limits exist as finite values.
↓
compare left and right
↓
decide the two-sided limit
\(x\to a^-\) means approach from values below \(a\), while \(x\to a^+\) means approach from values above \(a\). A finite two-sided limit exists when the two finite one-sided limits agree. The function value at \(a\) is separate from these limiting values, and at a domain boundary only one direction may be available.