Limits: Foundations and Properties in Business Calculus

Limits

Introduction to Limits

The concept of a limit is foundational in calculus, describing the behavior of a function as its input approaches a particular value. Limits allow us to analyze functions at points where they may not be explicitly defined, and are essential for understanding continuity, derivatives, and integrals.

• Limit Notation: The limit of a function f(x) as x approaches c is written as $\lim_{x \to c} f(x)$.

• Undefined Points: A function may be undefined at a point, but its limit as x approaches that point can still exist.

• Example: For $f(x) = \frac{x^2 - 4}{x - 2}$, f(2) is undefined, but as x approaches 2, f(x) approaches 4.

Numerical Exploration of Limits

Evaluating limits numerically involves examining the values of a function as x approaches a specific value from both sides.

• Approaching from the Left: As x approaches 2 from below, $f(x) = \frac{x^2 - 4}{x - 2}$ yields values increasingly close to 4.

• Approaching from the Right: As x approaches 2 from above, f(x) also approaches 4.

• Conclusion: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$.

Formal Definition of a Limit

The formal (epsilon-delta) definition quantifies the idea of a function approaching a value as x approaches c.

• We write $\lim_{x \to c} f(x) = L$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that whenever $0 < |x - c| < \delta$, it follows that $|f(x) - L| < \epsilon$.

• This means f(x) can be made arbitrarily close to L by taking x sufficiently close to c (but not equal to c).

• Additional info: While the epsilon-delta definition is rigorous, most business calculus courses focus on intuitive and algebraic approaches.

Limit Value vs. Function Value

The value of a function at a point and the limit as x approaches that point may differ.

• Example: For $f(x) = 1 + x$, $\lim_{x \to 3} f(x) = 4$ and $f(3) = 4$.

• Contrasting Example: For $g(x) = \begin{cases} 1 + x & x \neq 3 \\ 3 & x = 3 \end{cases}$, $\lim_{x \to 3} g(x) = 4$ but $g(3) = 3$.

• Key Point: The limit as x approaches c depends on the behavior near c, not necessarily the value at c.

One-Sided Limits

When the behavior of a function differs as x approaches c from the left and right, we use one-sided limits.

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

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

• Example: For $h(x) = \frac{|x|}{x}$, $\lim_{x \to 0^-} h(x) = -1$ and $\lim_{x \to 0^+} h(x) = 1$.

• Existence of Limit: The two-sided limit $\lim_{x \to c} f(x)$ exists if and only if both one-sided limits exist and are equal.

Algebraic Properties of Limits

Limits can be manipulated using algebraic rules, provided the individual limits exist.

• $\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$

• $\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$

• $\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

• $\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$ for constant k

• $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$ provided $\lim_{x \to c} g(x) \neq 0$

• $\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$ for integer n

• $\lim_{x \to c} \sqrt[p]{f(x)} = \sqrt[p]{\lim_{x \to c} f(x)}$ if the root is defined

Limits of Common Functions

• Constant Function: $\lim_{x \to c} k = k$

• Identity Function: $\lim_{x \to c} x = c$

• Power Function: $\lim_{x \to c} x^n = c^n$ for integer n

• Polynomial Function: If $f(x)$ is a polynomial, $\lim_{x \to c} f(x) = f(c)$

• Rational Function: If $f(x)$ and $g(x)$ are polynomials and $g(c) \neq 0$, $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{f(c)}{g(c)}$

Example: $\lim_{x \to 1} \frac{2x + 1}{x^2 + 1} = \frac{2 \cdot 1 + 1}{1^2 + 1} = \frac{3}{2}$

Limits by Simplification

When direct substitution yields an indeterminate form (such as 0/0), algebraic simplification can often resolve the limit.

• Factor and Cancel: Simplify the expression to eliminate the problematic factor.

• Example: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

  • Factor numerator: $x^2 - 4 = (x - 2)(x + 2)$

  • Cancel $(x - 2)$: $\frac{(x - 2)(x + 2)}{x - 2} = x + 2$ for $x \neq 2$

  • Take the limit: $\lim_{x \to 2} x + 2 = 4$

• Graphical Interpretation: The graph of $y = \frac{x^2 - 4}{x - 2}$ is the line $y = x + 2$ with a hole at $x = 2$.

General Principle for Simplification

If two functions are equal for all x near c (except possibly at x = c), then their limits as x approaches c are equal:

• $\text{If } f(x) = g(x) \text{ for all } x \neq c, \text{ then } \lim_{x \to c} f(x) = \lim_{x \to c} g(x)$

• This principle is especially useful for rational functions that yield indeterminate forms at x = c.

Summary Table: Limit Properties

Property	Formula	Condition
Sum	$\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$	Both limits exist
Difference	$\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$	Both limits exist
Product	$\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$	Both limits exist
Quotient	$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$	$\lim_{x \to c} g(x) \neq 0$
Power	$\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$	n integer, limit exists
Root	$\lim_{x \to c} \sqrt[p]{f(x)} = \sqrt[p]{\lim_{x \to c} f(x)}$	Root defined

Additional info: Mastery of limits is essential for understanding derivatives and integrals, which are central to business calculus applications such as marginal analysis, optimization, and area computations.