Limits and Derivatives: Core Concepts and Applications

Limits and Derivatives

2.1 The Tangent and Velocity Problems

The study of calculus begins with two fundamental problems: finding the tangent to a curve at a point and determining the instantaneous velocity of a moving object. Both problems lead to the concept of the derivative.

• Tangent Problem: Involves finding the slope of the tangent line to a curve at a specific point. This is equivalent to finding the instantaneous rate of change of the function at that point.

• Velocity Problem: Concerns finding the instantaneous velocity of an object, which is the rate of change of its position with respect to time.

• Connection: Both problems are solved using limits, which measure how a function behaves as its input approaches a particular value.

• Example: The slope of the tangent to the curve y = f(x) at x = a is given by the limit: $m = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

2.2 The Limit of a Function

The concept of a limit is foundational in calculus. It describes the value that a function approaches as the input approaches a certain point.

• Definition: The limit of f(x) as x approaches a is L if f(x) gets arbitrarily close to L as x approaches a from either side.

• Notation: $\lim_{x \to a} f(x) = L$

• One-Sided Limits: Limits can be taken from the left ($\lim_{x \to a^-} f(x)$) or right ($\lim_{x \to a^+} f(x)$).

• Example: $\lim_{x \to 2} (3x + 1) = 7$

2.3 Calculating Limits Using the Limit Laws

Limit laws provide rules for evaluating limits of functions, making calculations systematic and reliable.

• Sum Law: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

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

• Quotient Law: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$

• Example: $\lim_{x \to 3} (2x^2 - x) = 2(3^2) - 3 = 15$

2.4 The Precise Definition of a Limit

The formal (epsilon-delta) definition of a limit provides mathematical rigor to the concept of limits.

• Definition: $\lim_{x \to a} f(x) = L$ means that for every $\varepsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \varepsilon$.

• Purpose: Ensures that the function values can be made as close as desired to L by taking x sufficiently close to a.

• Example: Prove $\lim_{x \to 2} (3x + 1) = 7$ using the epsilon-delta definition.

2.5 Continuity

A function is continuous at a point if its limit at that point equals its value there. Continuity is essential for many calculus theorems and applications.

• Definition: f is continuous at a if $\lim_{x \to a} f(x) = f(a)$.

• Types of Discontinuity: Removable, jump, and infinite discontinuities.

• Example: The function f(x) = x^2 is continuous everywhere.

2.6 Limits at Infinity; Horizontal Asymptotes

Limits at infinity describe the behavior of functions as x grows without bound. Horizontal asymptotes are lines that the graph of a function approaches as x approaches infinity or negative infinity.

• Definition: $\lim_{x \to \infty} f(x) = L$ means f(x) approaches L as x increases without bound.

• Horizontal Asymptote: The line y = L is a horizontal asymptote if $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$.

• Example: $\lim_{x \to \infty} \frac{1}{x} = 0$; so y = 0 is a horizontal asymptote.

2.7 Derivatives and Rates of Change

The derivative measures the instantaneous rate of change of a function. It is a central concept in calculus, with applications in physics, engineering, and economics.

• Definition: The derivative of f at a is $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Interpretation: Represents the slope of the tangent line to the graph of f at x = a.

• Example: For f(x) = x^2, $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = 2x$

2.8 The Derivative as a Function

The derivative can be viewed as a new function derived from the original, assigning to each input the instantaneous rate of change at that point.

• Definition: The derivative function f' is defined by $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Notation: $f'(x)$, $\frac{df}{dx}$, $\frac{dy}{dx}$

• Example: If f(x) = x^3, then $f'(x) = 3x^2$