Premed · Premed · Calculus 1
Lecture 2: Limits — Intuitive and Formal Definitions
Calculus I — Differential Calculus
Learning Objectives
By the end of this lecture, students will be able to:
- Explain the intuitive meaning of a limit
- Estimate limits from graphs and tables of values
- Distinguish between left-hand and right-hand limits
- State the precise epsilon-delta definition of a limit
- Identify cases where limits fail to exist
- Understand the concept of limits at infinity and infinite limits
Lecture Content
I. The Idea of a Limit
The central question of this lecture is deceptively simple: what value does f(x) approach as x gets closer and closer to a particular number a? The notation lim_{x -> a} f(x) = L means "as x approaches a, f(x) approaches L." A critical distinction must be made right away: the limit describes the behavior near a, not necessarily the value at a. It is entirely possible for f(a) to be undefined while the limit still exists, and it is also possible for f(a) to exist but differ from the limit.
Consider the function f(x) = (x^2 - 1)/(x - 1), which is undefined at x = 1 due to division by zero. However, for any x other than 1, the expression can be simplified by factoring: (x^2 - 1)/(x - 1) = (x + 1)(x - 1)/(x - 1) = x + 1. As x approaches 1, x + 1 approaches 2, so lim_{x -> 1} f(x) = 2 even though f(1) does not exist.
II. Estimating Limits Numerically and Graphically
One practical way to estimate a limit is the numerical approach: construct a table of values with x approaching a from both sides and observe what happens. For instance, consider f(x) = (sin x)/x as x approaches 0. Evaluating at x = 0.1 gives f(x) = 0.9983..., and at x = 0.01 gives f(x) = 0.99998..., suggesting that lim_{x -> 0} (sin x)/x = 1.
The graphical approach involves tracing the curve from both sides to see whether the y-values converge to a single number. However, it is important to exercise caution: both numerical and graphical estimates can be misleading. They suggest a value for the limit but do not constitute a proof.
<image>Panel A: The graph of f(x) = (x^2 - 1)/(x - 1) showing the line y = x + 1 with a hollow circle (open dot) at the point (1, 2), indicating the function is undefined there but the limit exists. Panel B: A table of values showing x approaching 1 from the left (0.9, 0.99, 0.999) and from the right (1.1, 1.01, 1.001) with corresponding f(x) values converging to 2.</image>
III. One-Sided Limits
The left-hand limit lim_{x -> a^-} f(x) = L means that f(x) approaches L as x approaches a from the left (with x < a). The right-hand limit lim_{x -> a^+} f(x) = L means that f(x) approaches L as x approaches a from the right (with x > a). The two-sided limit lim_{x -> a} f(x) = L exists if and only if both one-sided limits exist and are equal: lim_{x -> a^-} f(x) = lim_{x -> a^+} f(x) = L.
A classic example where one-sided limits differ is the Heaviside step function H(x) = { 0 if x < 0; 1 if x >= 0 }. As x approaches 0 from the left, H(x) stays at 0, so the left-hand limit is 0. As x approaches 0 from the right, H(x) is 1, so the right-hand limit is 1. Since the two one-sided limits disagree, the two-sided limit at x = 0 does not exist.
<image>The graph of the Heaviside step function: a horizontal line at y = 0 for x < 0 (with an open dot at the origin), and a horizontal line at y = 1 for x >= 0 (with a closed dot at (0,1)). Arrows labeled "left-hand limit = 0" and "right-hand limit = 1" point to the respective values on the y-axis. Title: "One-sided limits need not agree."</image>
IV. When Limits Fail to Exist
A limit does not exist (DNE) in three main situations. First, the left-hand and right-hand limits may not be equal, as in a jump discontinuity. Second, the function may oscillate infinitely, as with lim_{x -> 0} sin(1/x), which bounces between -1 and 1 without ever settling on a single value. Third, the function may increase or decrease without bound, producing an infinite limit (discussed below). Being precise about why a limit does not exist is important: always state the reason.
V. Infinite Limits and Vertical Asymptotes
The notation lim_{x -> a} f(x) = +infinity means that f(x) grows without bound as x approaches a, while lim_{x -> a} f(x) = -infinity means f(x) decreases without bound. Strictly speaking, these limits "do not exist" as real numbers, but the infinity notation is used to describe the behavior precisely.
A vertical line x = a is called a vertical asymptote of f if at least one of the one-sided limits at a is +infinity or -infinity. For example, f(x) = 1/(x - 3)^2 satisfies lim_{x -> 3} f(x) = +infinity, so x = 3 is a vertical asymptote.
VI. Limits at Infinity and Horizontal Asymptotes
The notation lim_{x -> infinity} f(x) = L means that f(x) approaches L as x grows without bound to the right, and lim_{x -> -infinity} f(x) = L means that f(x) approaches L as x goes to negative infinity. A horizontal line y = L is a horizontal asymptote of the graph if either of these limits equals L.
Several key results govern limits at infinity. For any positive integer n, lim_{x -> infinity} 1/x^n = 0. For rational functions P(x)/Q(x), the behavior depends on comparing the degrees of P and Q. If the degree of P is less than the degree of Q, the limit is 0. If the degrees are equal, the limit is the ratio of the leading coefficients. If the degree of P exceeds the degree of Q, the limit is plus or minus infinity.
VII. The Precise (Epsilon-Delta) Definition
The formal definition of a limit states: lim_{x -> a} f(x) = L means that for every epsilon > 0, there exists a delta > 0 such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon.
The idea behind this definition is that epsilon represents the tolerance on the output (how close f(x) must be to L), while delta represents the required proximity of x to a. The definition asserts that no matter how tight the output tolerance, we can always find an input tolerance that works. The condition 0 < |x - a| excludes x = a itself, reflecting the fact that we only care about the behavior near a, not at a. This definition makes rigorous what "approaches" means and provides the foundation for all of calculus.
<image>An epsilon-delta diagram: a graph of a generic increasing function y = f(x) with a point (a, L) highlighted. A horizontal band of width 2epsilon is drawn around y = L (shaded in light blue). A vertical band of width 2delta is drawn around x = a (shaded in light yellow). The portion of the curve within the vertical band is shown to remain inside the horizontal band. Labels: "epsilon" on the y-axis bands, "delta" on the x-axis bands, with arrows indicating the widths. Caption: "For every epsilon-band around L, there exists a delta-band around a that keeps f(x) inside the epsilon-band."</image>
