Limits and One-Sided Limits in Calculus: Definitions, Properties, and Examples

Limits in Calculus

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as their inputs approach specific values. Understanding limits is essential for analyzing continuity, derivatives, and integrals.

• Limit of a function: The value that a function approaches as the input approaches a particular point.

• Undefined points: Some functions may be undefined at certain points, but their limits may still exist as the input nears those points.

Example Functions and Their Limits

Consider the following functions:

• $f(x) = \frac{x^2 - 4}{x - 2}$

• $g(x) = \frac{|x - 2|}{x - 2}$

Domain: Both functions are undefined at $x = 2$.

• For $f(x)$: $f(2)$ is undefined because the denominator is zero.

• For $g(x)$: $g(2)$ is also undefined for the same reason.

To analyze their behavior near $x = 2$:

• $\lim_{x \to 2^+} g(x) = \lim_{x \to 2^+} \frac{|x - 2|}{x - 2} = 1$

• $\lim_{x \to 2^-} g(x) = \lim_{x \to 2^-} \frac{|x - 2|}{x - 2} = -1$

Key Point: The behavior of a function near a point can be different from its value at that point.

One-Sided Limits

Definition and Notation

One-sided limits describe the behavior of a function as the input approaches a point from one direction only.

• Right-hand limit: $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right ($x > a$).

• Left-hand limit: $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left ($x < a$).

If both one-sided limits exist and are equal, then the two-sided limit exists:

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

Graphical Representation

Graphs can illustrate the difference between left-hand and right-hand limits, especially at points of discontinuity.

• If $f(x)$ approaches different values from the left and right, the two-sided limit does not exist at that point.

Absolute Value Functions and Limits

Definition of Absolute Value

The absolute value of a real number $a$ is defined as:

• $|a| = a$ if $a \geq 0$

• $|a| = -a$ if $a < 0$

For expressions involving absolute value, such as $|x - 1|$:

• $|x - 1| = x - 1$ if $x \geq 1$

• $|x - 1| = -(x - 1)$ if $x < 1$

Evaluating Limits with Absolute Value

When evaluating limits involving absolute value, consider the direction from which $x$ approaches the point.

• For $\lim_{x \to 3^-} \frac{|x^2 - 9|}{x - 3}$, since $x < 3$, $x^2 - 9 < 0$, so $|x^2 - 9| = -(x^2 - 9)$.

• For $\lim_{x \to 3^+} \frac{|x^2 - 9|}{x - 3}$, since $x > 3$, $x^2 - 9 > 0$, so $|x^2 - 9| = x^2 - 9$.

Example:

• $\lim_{x \to 3^-} \frac{|x^2 - 9|}{x - 3} = \lim_{x \to 3^-} \frac{-(x^2 - 9)}{x - 3} = \lim_{x \to 3^-} \frac{-(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3^-} -(x + 3) = -6$

• $\lim_{x \to 3^+} \frac{|x^2 - 9|}{x - 3} = \lim_{x \to 3^+} \frac{(x^2 - 9)}{x - 3} = \lim_{x \to 3^+} (x + 3) = 6$

Since the left and right limits are not equal, the two-sided limit does not exist:

• $\lim_{x \to 3} \frac{|x^2 - 9|}{x - 3}$ does not exist (DNE).

Infinite Limits

Types of Infinite Limits

Infinite limits describe the behavior of functions as they increase or decrease without bound near a particular point.

• Infinite limits from the left: If $f(x)$ increases without bound as $x \to a^-$, then $\lim_{x \to a^-} f(x) = +\infty$. If $f(x)$ decreases without bound, $\lim_{x \to a^-} f(x) = -\infty$.

• Infinite limits from the right: If $f(x)$ increases without bound as $x \to a^+$, then $\lim_{x \to a^+} f(x) = +\infty$. If $f(x)$ decreases without bound, $\lim_{x \to a^+} f(x) = -\infty$.

• Two-sided infinite limits: If $f(x)$ increases without bound as $x \to a$, then $\lim_{x \to a} f(x) = +\infty$. If $f(x)$ decreases without bound, $\lim_{x \to a} f(x) = -\infty$.

Type	Condition	Limit Notation	Result
Infinite limit from left	$x \to a^-$, $f(x)$ increases without bound	$\lim_{x \to a^-} f(x)$	$+\infty$
Infinite limit from left	$x \to a^-$, $f(x)$ decreases without bound	$\lim_{x \to a^-} f(x)$	$-\infty$
Infinite limit from right	$x \to a^+$, $f(x)$ increases without bound	$\lim_{x \to a^+} f(x)$	$+\infty$
Infinite limit from right	$x \to a^+$, $f(x)$ decreases without bound	$\lim_{x \to a^+} f(x)$	$-\infty$
Two-sided infinite limit	$x \to a$, $f(x)$ increases without bound	$\lim_{x \to a} f(x)$	$+\infty$
Two-sided infinite limit	$x \to a$, $f(x)$ decreases without bound	$\lim_{x \to a} f(x)$	$-\infty$

Graphical Examples of Infinite Limits

Graphs can show functions approaching infinity or negative infinity as $x$ approaches a specific value from the left or right.

• For $y = f(x)$, $\lim_{x \to a^+} f(x) = +\infty$

• For $y = g(x)$, $\lim_{x \to a^+} g(x) = -\infty$

• For $y = h(x)$, $\lim_{x \to a^-} h(x) = +\infty$

• For $y = k(x)$, $\lim_{x \to a^-} k(x) = -\infty$

Summary Table: Types of Limits

Type of Limit	Notation	Description
Two-sided limit	$\lim_{x \to a} f(x)$	Approaches $a$ from both sides
Right-hand limit	$\lim_{x \to a^+} f(x)$	Approaches $a$ from the right ($x > a$)
Left-hand limit	$\lim_{x \to a^-} f(x)$	Approaches $a$ from the left ($x < a$)
Infinite limit	$\lim_{x \to a} f(x) = \pm\infty$	Function increases or decreases without bound

Key Takeaways

• Limits describe the behavior of functions near specific points, even if the function is undefined at those points.

• One-sided limits are crucial for analyzing discontinuities and piecewise functions.

• Absolute value functions require careful consideration of the direction of approach.

• Infinite limits indicate unbounded behavior near certain points.