One-Sided Limits and Limits at Infinity - Study Notes

Notation and core ideas

One-sided limits are written with a superscript indicating the direction of approach. If f(x) has a right-hand limit L as x approaches x0, we write
\lim{x \to x0^+} f(x) = L.
If f(x) has a left-hand limit L as x approaches x0, we write
\lim{x \to x0^-} f(x) = L.
A two-sided limit at x0 exists only if both one-sided limits exist and are equal:
\lim{x \to x0} f(x) = L \quad\text{iff}\quad \lim{x \to x0^+} f(x) = \lim{x \to x0^-} f(x) = L.
If the two one-sided limits exist but are different, the two-sided limit does not exist.

The examples below illustrate how to compute one-sided limits by using the expression valid on the relevant side, and how to reason about the existence of the two-sided limit from the one-sided limits.

Example setup: a potential singularity at x = -2

Consider
f(x)=\frac{(x+3)|x+2|}{x+2}.
The denominator is zero at $x=-2$, so we examine the one-sided limits as $x\to -2$ from the left and from the right.

Example 1: Absolute value near $x=-2$ (left-hand and right-hand limits)

For $x<-2$, we have $x+2<0$ and hence $|x+2|=-(x+2)$. Thus f(x)=\frac{(x+3)|x+2|}{x+2}=\frac{(x+3)(-(x+2))}{x+2}=-(x+3). Therefore \lim{x\to -2^-} f(x)=\lim{x\to -2^-} (-(x+3))= -((-2)+3) = -1. For $x>-2$, we have $x+2>0$ and hence $|x+2|=x+2$. Thus
f(x)=\frac{(x+3)|x+2|}{x+2}=\frac{(x+3)(x+2)}{x+2}=x+3,
so
\lim{x\to -2^+} f(x)=\lim{x\to -2^+} (x+3)=(-2)+3=1.
Since $-1\neq 1$, the two-sided limit does not exist:
\lim_{x\to -2} f(x) \quad\text{DNE}.

Example 2: Piecewise linear approach to $x=2$ (one-sided vs two-sided)

Define
f(x)=\begin{cases}3-x,& x<2,\ x/2+1,& x>2.
\end{cases}
The left-hand limit uses the expression valid to the left of 2:
\lim{x\to 2^-} f(x)=\lim{x\to 2^-} (3-x)=3-2=1.
The right-hand limit uses the expression valid to the right of 2:
\lim{x\to 2^+} f(x)=\lim{x\to 2^+} \left(\frac{x}{2}+1\right)=\frac{2}{2}+1=2.
Thus the two one-sided limits exist but are not equal, so
\lim_{x\to 2} f(x)\text{ DNE}.

Example 3: Square root expression near $x=-0.5$ (one-sided becoming two-sided)

Consider
g(x)=\sqrt{\frac{x+2}{x+1}}.
At $x=-0.5$, the denominator is $x+1=0.5\neq 0$ and the radicand is positive near there, so the expression is defined in a neighborhood of $-0.5$ on both sides. The left-hand limit (which, in this case, agrees with the right-hand limit) is
\lim{x\to -0.5} g(x)=\sqrt{\frac{-0.5+2}{-0.5+1}}=\sqrt{\frac{1.5}{0.5}}=\sqrt{3}. Since the function is well-defined around $-0.5$ from both sides and the limit yields the same value, both the left-hand and right-hand limits exist and equal the two-sided limit: \lim{x\to -0.5} g(x)=\sqrt{3}.

Example 4: A modified cubic with a removable value at $x=1$

Let
f(x)=\begin{cases}x^3,& x\neq 1,\ 0,& x=1.
\end{cases}
Graphically this is the cubic $y=x^3$ with the point $(1,1)$ removed and replaced by a hole, filled at $(1,0)$.
To compute the one-sided limits, note that for $x<1$ (and for $x>1$ but near 1) we have $f(x)=x^3$ since $x\neq 1$ in these neighborhoods. Therefore,
\lim{x\to 1^-} f(x)=\lim{x\to 1^-} x^3=1,
\lim{x\to 1^+} f(x)=\lim{x\to 1^+} x^3=1.
Because both one-sided limits exist and are equal to 1, the two-sided limit exists and equals 1:
\lim_{x\to 1} f(x)=1.

Takeaways and connections

  • One-sided limits are evaluated using the expression valid on the corresponding side of the point where the limit is taken.

  • If a two-sided limit exists, both one-sided limits exist and are equal to the same value.

  • If the two one-sided limits exist but differ, the two-sided limit does not exist.

  • If the expression is well-defined in a neighborhood around the point (even if the function is redefined at the point), the two-sided limit can exist and equal the common value of the one-sided limits.

  • In cases involving absolute value, the sign of the inner expression (e.g., whether x+2 is positive or negative near the point) can flip the sign of the expression, leading to different one-sided limits.

  • In piecewise definitions, examine each piece on its domain side to determine the corresponding one-sided limit.

  • In cases where a function equals a simple expression on both sides (e.g., a polynomial like $x^3$) and the limit point is not a region where the expression changes, the one-sided limits often reduce to the same two-sided limit.

Limits at infinity (note)

The transcript focus is on one-sided limits. If needed, a separate study note would cover limits at infinity, including definitions like $\lim{x\to\infty} f(x)$ and $\lim{x\to -\infty} f(x)$, and common techniques for evaluating them.

Notation and core ideas

One-sided limits are fundamental to understanding the behavior of a function around a specific point. They describe the value a function approaches as the input approaches a certain point from either the left or the right side.

1. Right-hand limit

  • Definition: If a function f(x) approaches a specific value L as x approaches x0 from values greater than x0 (i.e., from the right side), then L is called the right-hand limit of f(x) at x_0.

  • Notation: This is mathematically expressed as:
    \lim{x \to x0^+} f(x) = L

  • Interpretation: As x gets arbitrarily close to x0 while remaining larger than x0, the corresponding values of f(x) get arbitrarily close to L.

2. Left-hand limit

  • Definition: If a function f(x) approaches a specific value L as x approaches x0 from values smaller than x0 (i.e., from the left side), then L is called the left-hand limit of f(x) at x_0.

  • Notation: This is mathematically expressed as:
    \lim{x \to x0^-} f(x) = L

  • Interpretation: As x gets arbitrarily close to x0 while remaining smaller than x0, the corresponding values of f(x) get arbitrarily close to L.

3. Two-sided limit

  • Definition: A two-sided limit of f(x) at a point x0 exists if and only if both the left-hand limit and the right-hand limit at x0 exist and are equal to the same value.

  • Notation and Condition: The existence of a two-sided limit L at x0 is formally stated as: \lim{x \to x0} f(x) = L \quad\text{iff}\quad \lim{x \to x0^+} f(x) = \lim{x \to x_0^-} f(x) = L.

  • Crucial Implication: If the two one-sided limits exist but have different values (i.e., L1 \neq L2), then the two-sided limit at that point does not exist. This is explicitly demonstrated in examples where functions like absolute values or piecewise definitions cause the function to approach different values from different directions