One-Sided Limits in Detail

Think of it as: "What happens as we walk toward a from the left side of the number line?"

Key Points:

• The superscript - means "from below" or "from the left"
• We only look at x-values less than a
• Example: approaching x = 3 from values like 2.9, 2.99, 2.999...

Right-Hand Limit

The right-hand limit examines the function as we approach from the right (larger values):

$\lim_{x \to a^+} f(x) = L$

Think of it as: "What happens as we walk toward a from the right side of the number line?"

Key Points:

• The superscript + means "from above" or "from the right"
• We only look at x-values greater than a
• Example: approaching x = 3 from values like 3.1, 3.01, 3.001...

The Big Rule

For a two-sided limit to exist, both one-sided limits must exist AND be equal!

$\lim_{x \to a} f(x) = L \text{ if and only if } \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

Example: When One-Sided Limits Differ

Consider a piecewise function:

$f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ x + 5 & \text{if } x \geq 2 \end{cases}$

From the left (using $x + 1$): $\lim_{x \to 2^-} f(x) = 2 + 1 = 3$

From the right (using $x + 5$): $\lim_{x \to 2^+} f(x) = 2 + 5 = 7$

Since $3 \neq 7$, the two-sided limit does not exist!

$\lim_{x \to 2} f(x) = \text{DNE}$

Why This Matters

One-sided limits are crucial for:

• Piecewise functions that change formula at a point
• Functions with jumps (discontinuities)
• Rational functions near vertical asymptotes
• Absolute value functions at the vertex

Visual Interpretation

On a graph:

• Left-hand limit: Cover everything to the right of a, look where the curve is heading
• Right-hand limit: Cover everything to the left of a, look where the curve is heading
• If they point to different heights, the two-sided limit DNE

Left-hand limit (x < 5):

$\lim_{x \to 5^-} \frac{|x - 5|}{x - 5} = \lim_{x \to 5^-} \frac{5 - x}{x - 5} = \lim_{x \to 5^-} \frac{-(x - 5)}{x - 5} = \lim_{x \to 5^-} (-1) = -1$

Right-hand limit (x ≥ 5):

$\lim_{x \to 5^+} \frac{|x - 5|}{x - 5} = \lim_{x \to 5^+} \frac{x - 5}{x - 5} = \lim_{x \to 5^+} 1 = 1$

Result:

• Left-hand limit: -1
• Right-hand limit: 1
• Two-sided limit: DNE (because -1 ≠ 1)