Limits and Interval Notation: Foundations of Calculus

Interval Notation and Real Numbers

Intervals on the Real Line

In calculus, intervals are used to describe sets of real numbers between two endpoints. Understanding interval notation is essential for discussing domains, ranges, and the behavior of functions.

• Open Interval: $(a, b) = \{x \in \mathbb{R} : a < x < b\}$ All real numbers strictly between $a$ and $b$.

• Closed Interval: $[a, b] = \{x \in \mathbb{R} : a \leq x \leq b\}$ All real numbers between $a$ and $b$, including the endpoints.

• Half-Open (Half-Closed) Intervals:

  • $[a, b) = \{x \in \mathbb{R} : a \leq x < b\}$

  • $(a, b] = \{x \in \mathbb{R} : a < x \leq b\}$

  These include one endpoint but not the other.

• Rays:

  • $(a, \infty) = \{x \in \mathbb{R} : x > a\}$ (open ray to the right)

  • $[a, \infty) = \{x \in \mathbb{R} : x \geq a\}$ (closed ray to the right)

  • $(-\infty, a) = \{x \in \mathbb{R} : x < a\}$ (open ray to the left)

  • $(-\infty, a] = \{x \in \mathbb{R} : x \leq a\}$ (closed ray to the left)

Key Term: The symbol "$\in$" means "is an element of" or "is in".

Limits: A Fundamental Concept

Intuitive Definition of a Limit

The concept of a limit describes the behavior of a function as its input approaches a particular value. Limits are foundational for defining continuity, derivatives, and integrals.

• Definition: Let $f(x)$ be a function defined on an interval around $a$ (except possibly at $a$ itself). We say the limit of $f(x)$ as $x$ approaches $a$ is $L$ if $f(x)$ can be made arbitrarily close to $L$ for all $x$ sufficiently close to $a$ (but not equal to $a$).

This is denoted as:

• $\lim_{x \to a} f(x) = L$

• or $f(x) \to L$ as $x \to a$

• If $f(x)$ does not approach any real number $L$ as $x$ approaches $a$, we say the limit does not exist (DNE).

Key Point: "Arbitrarily close" means as close as desired. If $f(x)$ can be made as close to $L$ as we want by choosing $x$ sufficiently close to $a$, then the limit exists.

Examples of Limits

• Example 1: $f(x) = |x|$. Find $\lim_{x \to 0} f(x)$.

  • As $x$$approaches $0$|x|$approaches $0$. Thus, $\lim_{x \to 0} |x| = 0$.

• Example 2: $f(x) = \begin{cases} x-1 & x \geq 0 \\ x+1 & x < 0 \end{cases}$, find $\lim_{x \to 0} f(x)$.

  • Left-hand limit: $x \to 0^-$, $f(x) \to 1$.

  • Right-hand limit: $x \to 0^+$, $f(x) \to -1$.

  • Since the left and right limits are not equal, $\lim_{x \to 0} f(x)$ does not exist.

• Example 3: $f(x) = \frac{\sin(x)}{x}$, find $\lim_{x \to 0} \frac{\sin(x)}{x}$.

  • By graphing or using the Squeeze Theorem, $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.

One-Sided Limits

Left-Hand and Right-Hand Limits

One-sided limits describe the behavior of a function as $x$ approaches a value from only one side.

• Left-Hand Limit: $\lim_{x \to a^-} f(x) = L_1$ means $f(x)$ approaches $L_1$ as $x$ approaches $a$ from the left ($x < a$).

• Right-Hand Limit: $\lim_{x \to a^+} f(x) = L_2$ means $f(x)$ approaches $L_2$ as $x$ approaches $a$ from the right ($x > a$).

If both one-sided limits exist and are equal, the (two-sided) limit exists and equals this common value.

Example: Piecewise Function

Consider the function:

• $f(x) = \begin{cases} 2\sin(x) & 0 \leq x < \pi \\ \frac{x}{\pi} + 1 & \pi \leq x < 3\pi \end{cases}$

Calculate the following, if they exist:

• $f(\pi)$

• $\lim_{x \to \pi^-} f(x)$

• $\lim_{x \to \pi^+} f(x)$

• $f(3\pi)$

• $\lim_{x \to 3\pi^-} f(x)$

• $\lim_{x \to 3\pi^+} f(x)$

Application: This type of analysis is essential for understanding continuity and discontinuities in piecewise-defined functions.

Formal Definition of a Limit

Two-Sided Limit

Let $f : D \to \mathbb{R}$ be a function and $a \in \mathbb{R}$. If both $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ exist and are equal to $L$, then the limit exists and is written as:

• $\lim_{x \to a} f(x) = L$

If the left and right limits are not equal, the limit does not exist (DNE).

Summary Table: Types of Limits

Type	Notation	Description
Two-sided limit	$\lim_{x \to a} f(x)$	Approaches $a$ from both sides
Left-hand limit	$\lim_{x \to a^-} f(x)$	Approaches $a$ from the left
Right-hand limit	$\lim_{x \to a^+} f(x)$	Approaches $a$ from the right

Key Takeaways

• Interval notation is used to describe sets of real numbers and is foundational for calculus.

• The limit of a function describes its behavior as the input approaches a specific value.

• One-sided limits help analyze functions that behave differently from the left and right.

• For a limit to exist at a point, both one-sided limits must exist and be equal.

• Piecewise functions require careful analysis of limits from both sides at points where the formula changes.

Example Application: Limits are used to define continuity, derivatives, and integrals, which are the core concepts of calculus.