Limits and Continuity: Foundations of Calculus

Limits and Continuity

Introduction to Limits

The concept of a limit is the cornerstone of calculus, providing a rigorous way to describe how functions behave as their inputs approach specific values. Limits allow us to define instantaneous rates of change (such as velocity and acceleration) and the area under curves, which are foundational to differential and integral calculus.

• Instantaneous Rate of Change: Calculus formalizes the idea of how a quantity changes at an exact moment, such as the velocity of a skydiver as they fall through the air. Air resistance prevents the velocity from increasing indefinitely; instead, it approaches a terminal velocity, a limiting value.

• Geometric Problems: Two classic problems led to the development of calculus:

  • The Tangent Line Problem: Find the equation of the tangent to a curve at a point.

  • The Area Problem: Find the area under a curve between two points.

Intuitive Notion of Limits

Informally, the limit of a function f(x) as x approaches a is the value that f(x) gets closer to as x gets closer to a (but not necessarily at a itself). This is written as:

• $\lim_{x \to a} f(x) = L$

Key points:

• The value of f(x) at x = a does not affect the limit.

• Limits can be estimated numerically (tables of values) or graphically.

One-Sided Limits

Sometimes, the behavior of a function as x approaches a from the left ($x \to a^-$) or right ($x \to a^+$) is different. These are called one-sided limits:

• $\lim_{x \to a^-} f(x)$: Limit from the left

• $\lim_{x \to a^+} f(x)$: Limit from the right

The two-sided limit exists if and only if both one-sided limits exist and are equal.

Infinite Limits and Vertical Asymptotes

If the values of f(x) increase or decrease without bound as x approaches a, we write:

• $\lim_{x \to a} f(x) = +\infty$ or $-\infty$

This behavior is associated with vertical asymptotes at $x = a$.

Computing Limits Algebraically

Several theorems allow us to compute limits efficiently:

• Sum, Difference, Product, and Quotient Laws: The limit of a sum is the sum of the limits, and similarly for differences, products, and quotients (provided the denominator's limit is not zero).

• Limits of Polynomials: $\lim_{x \to a} p(x) = p(a)$ for any polynomial $p(x)$.

• Limits of Rational Functions: If $q(a) \neq 0$, $\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}$.

• Indeterminate Forms: If both numerator and denominator approach zero, algebraic simplification (such as factoring or rationalizing) is needed.

Limits at Infinity and End Behavior

Limits can also describe the behavior of functions as x increases or decreases without bound ($x \to +\infty$ or $x \to -\infty$):

• Horizontal Asymptotes: If $\lim_{x \to \pm\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

• End Behavior of Polynomials: The highest degree term dominates as $x \to \pm\infty$.

• End Behavior of Rational Functions: Compare the degrees of numerator and denominator to determine the limit at infinity.

Rigorous Definition of a Limit (Epsilon-Delta)

The formal definition of a limit uses two parameters:

• For every $\epsilon > 0$, there exists $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \epsilon$.

This definition ensures that $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ (but not equal to $a$).

Continuity

A function is continuous at $x = c$ if:

• $f(c)$ is defined

• $\lim_{x \to c} f(x)$ exists

• $\lim_{x \to c} f(x) = f(c)$

Types of discontinuities:

• Removable: Limit exists, but $f(c)$ is not defined or not equal to the limit.

• Jump: One-sided limits exist but are not equal.

• Infinite: Function increases or decreases without bound near $c$.

In real-world applications, continuity models unbroken phenomena, such as the path of a thrown baseball.

Continuity in Applications

Discontinuities often signal important events, such as a sudden drop in voltage when a cable is cut.

Properties of Continuous Functions

• Polynomials are continuous everywhere.

• Rational functions are continuous wherever the denominator is nonzero.

• The sum, difference, product, and (where defined) quotient of continuous functions are continuous.

• The composition of continuous functions is continuous.

The Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists $c$ in $[a, b]$ such that $f(c) = k$. This theorem is fundamental for proving the existence of roots and for numerical approximation methods.

Continuity of Trigonometric, Exponential, and Inverse Functions

• Trigonometric functions (sin, cos, tan, etc.) are continuous on their natural domains.

• Exponential functions $b^x$ are continuous for all real $x$; logarithmic functions $\log_b x$ are continuous for $x > 0$.

• The inverse of a continuous one-to-one function is continuous on its domain.

The Squeezing Theorem

If $g(x) \leq f(x) \leq h(x)$ near $c$, and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$. This is especially useful for evaluating tricky limits, such as $\lim_{x \to 0} \frac{\sin x}{x} = 1$.

Summary Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, $f(c)$ undefined or not equal to limit	$f(x) = \frac{x^2-1}{x-1}$ at $x=1$
Jump	One-sided limits exist, not equal	$f(x) = \begin{cases}1, x<0 \\ 2, x\geq 0\end{cases}$ at $x=0$
Infinite	Function increases/decreases without bound	$f(x) = \frac{1}{x}$ at $x=0$

Key Formulas

• $\lim_{x \to a} k = k$

• $\lim_{x \to a} x = a$

• $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$

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

• $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if denominator's limit is not zero

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Example: Finding a Tangent Line Using Limits

To find the tangent to $y = x^2$ at $x = 1$:

• Slope of secant: $m_{sec} = \frac{x^2 - 1}{x - 1} = x + 1$ (for $x \neq 1$)

• Slope of tangent: $m_{tan} = \lim_{x \to 1} m_{sec} = 2$

• Equation: $y - 1 = 2(x - 1)$ or $y = 2x - 1$

Example: Approximating Roots with IVT

If $f(1) = -1$ and $f(2) = 5$, and $f$ is continuous, then $f(x) = 0$ has a solution in $(1,2)$.

Additional info: This guide covers the foundational concepts of limits and continuity, including intuitive and formal definitions, computational techniques, and their importance in calculus and real-world applications.