Constant Functions | Tony's Intertwingled Memex

Core Definition

A constant function is a function of the form $f(x) = c$ , where $c \in \mathbb{R}$ is a fixed real number. The function assigns the exact same output $c$ to every input $x$ in its domain. Its limit as $x$ approaches any value $a$ is simply the constant itself:

\lim_{x \to a} c = c

Concept Explanation

Graphically, a constant function forms a perfectly horizontal line at height $y = c$ . Because the function's height never changes:

As $x$ approaches any coordinate $a$ from the left ( $x \to a^-$ ), the height is $c$ .
As $x$ approaches $a$ from the right ( $x \to a^+$ ), the height is $c$ .
Both directional paths are constant and agree everywhere, meaning the limit is always determinate and continuous on the entire real line $\mathbb{R}$ .

        y ^
          |
        c +------------------- y = c
          |         |
          +---------+---------> x
                    a

Why We Use It in Calculus

Baseline for Limit Laws: Serves as the building block for the Constant Multiple Law and polynomial term evaluation.
Piecewise Boundaries: Many engineering piecewise models use constant values to represent inactive states, baseline voltages, or reset thresholds (e.g. $f(x) = 0$ when $x < 0$ ).
Derivatives & Rates of Change: Because constant functions do not change height, their rate of change (derivative) is identically zero: $\frac{d}{dx}[c] = 0$ .

Example Problems

Example Problem 1: Basic Evaluation of Constant Limits

Evaluate the following limits: (a) $\lim_{x \to 5} 17$
(b) $\lim_{x \to -3} \pi$
(c) $\lim_{x \to \infty} (-4.5)$

Solution

Since the expression inside each limit does not depend on $x$ , apply the constant rule $\lim_{x \to a} c = c$ :

(a) $\lim_{x \to 5} 17 = 17$
(b) $\lim_{x \to -3} \pi = \pi$
(c) $\lim_{x \to \infty} (-4.5) = -4.5$

Example Problem 2: Constant Branch in a Piecewise Function

Consider the piecewise function:

f(x) = \begin{cases} 4 & : x < 2 \\ x^2 & : x \ge 2 \end{cases}

Evaluate

\lim_{x \to 2^-} f(x)

and

\lim_{x \to 2^+} f(x)

, and determine whether

\lim_{x \to 2} f(x)

exists.

Solution

Step 1: Evaluate the left-sided limit ( $x < 2$ )
Approaching $2$ from the left uses the constant branch:

\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (4) = 4

Step 2: Evaluate the right-sided limit ( $x > 2$ )
Approaching $2$ from the right uses the quadratic branch:

\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (x^2) = 2^2 = 4

Step 3: Compare directional limits
Since $\lim_{x \to 2^-} f(x) = 4 = \lim_{x \to 2^+} f(x)$ , by the Existence Theorem, the overall limit exists:

\lim_{x \to 2} f(x) = 4

Example Problem 3: Reference Voltage Baseline (Electrical Engineering)

A digital logic gate outputs a constant high voltage level $V_{\text{high}}(t) = 5.0\text{ V}$ during standby mode. What is the limit of the logic gate output as time $t$ approaches the clock transition threshold $t \to 10\text{ ns}$ ?

Solution

Since the gate remains at a steady DC state prior to the clock pulse:

\lim_{t \to 10} V_{\text{high}}(t) = \lim_{t \to 10} (5.0\text{ V}) = 5.0\text{ V}

Answer: $5.0\text{ V}$

Connections & References

Parent Concepts: Limits, Calculus Foundations
Related Concepts: Identity Functions, Constant Multiple Law, Piecewise Functions
Course Links: Calc-1 Session 03