Calculus I - Week 1
Limits and Continuity
This week introduces the foundation of calculus: limits. Students learn how to evaluate limits numerically, graphically, and algebraically, then use limits to understand continuity and the beginning of instantaneous motion.
Learning Objectives
- Explain limits using plain language and mathematical notation. - Evaluate limits by direct substitution. - Recognize indeterminate forms like 0/0. - Use factoring to evaluate limits. - Use conjugates to simplify limits involving radicals. - Understand left-hand and right-hand limits. - Determine whether a two-sided limit exists. - Identify removable, jump, and infinite discontinuities. - Determine whether a function is continuous at a point. - Connect limits to instantaneous velocity in physics.
1. The Big Idea of a Limit
A limit asks what a function approaches as x gets close to a certain value. The word approaches is the key. We are not always asking what the function equals at that point. We are asking what the y-values get close to as x moves toward that point. Example: If f(x) = x + 2, then as x approaches 3, f(x) approaches 5. We write: lim x->3 f(x) = 5 This means the y-values get closer and closer to 5 as x gets closer and closer to 3.
2. Why Limits Matter in Calculus
Calculus is built on change. To understand change at one exact instant, we need limits. A derivative is a limit. Instantaneous velocity is a limit. The slope of a curve at one exact point is a limit. Without limits, we can only talk about average change over an interval. With limits, we can talk about what happens at a single instant.
3. Evaluating Limits by Direct Substitution
The first method is direct substitution. If plugging in the x-value gives a normal number, that number is the limit. Example 1: Evaluate lim x->4 (2x + 1). Substitute x = 4: 2(4) + 1 = 9 So: lim x->4 (2x + 1) = 9 Example 2: Evaluate lim x->-1 (x^2 + 3x). Substitute x = -1: (-1)^2 + 3(-1) = 1 - 3 = -2 So the limit is -2.
4. When Direct Substitution Gives 0/0
Sometimes direct substitution gives 0/0. This does not mean the answer is 0. It also does not mean the limit automatically does not exist. 0/0 is called an indeterminate form. It means more work is needed. Most early Calculus I limit problems that produce 0/0 can be solved by simplifying the expression first.
5. Factoring Limits
Example: Evaluate lim x->2 (x^2 - 4)/(x - 2). Direct substitution gives: (4 - 4)/(2 - 2) = 0/0 Factor the numerator: x^2 - 4 = (x - 2)(x + 2) Rewrite: ((x - 2)(x + 2))/(x - 2) Cancel the common factor: x + 2 Now substitute x = 2: 2 + 2 = 4 So the limit is 4. Important: We are not saying the original function is defined at x = 2. We are saying the function approaches 4 as x gets close to 2.
6. More Factoring Examples
Example: Evaluate lim x->3 (x^2 - 9)/(x - 3). Factor: x^2 - 9 = (x - 3)(x + 3) Cancel: ((x - 3)(x + 3))/(x - 3) = x + 3 Substitute: 3 + 3 = 6 So the limit is 6. Example: Evaluate lim x->-2 (x^2 - 4)/(x + 2). Factor: x^2 - 4 = (x - 2)(x + 2) Cancel: (x - 2)(x + 2)/(x + 2) = x - 2 Substitute x = -2: -2 - 2 = -4 So the limit is -4.
7. Limits with Radicals and Conjugates
Some limits involving square roots can be simplified using a conjugate. The conjugate of sqrt(x + 1) - 2 is sqrt(x + 1) + 2. Example: Evaluate lim x->3 (sqrt(x + 1) - 2)/(x - 3). Direct substitution gives 0/0. Multiply top and bottom by the conjugate: (sqrt(x + 1) + 2)/(sqrt(x + 1) + 2) The numerator becomes: (x + 1) - 4 = x - 3 Now cancel x - 3. The expression becomes: 1/(sqrt(x + 1) + 2) Substitute x = 3: 1/(sqrt(4) + 2) = 1/(2 + 2) = 1/4 So the limit is 1/4.
8. One-Sided Limits
A left-hand limit looks at what happens as x approaches a value from the left. A right-hand limit looks at what happens as x approaches a value from the right. If both sides approach the same number, the two-sided limit exists. If the left-hand and right-hand limits approach different numbers, the two-sided limit does not exist.
9. Continuity
A function is continuous at x = a when three things are true: 1. f(a) exists. 2. lim x->a f(x) exists. 3. lim x->a f(x) = f(a). In plain English, the graph has no break, hole, jump, or vertical asymptote at that point. A hole is often called a removable discontinuity. A sudden jump is called a jump discontinuity. A vertical blow-up is called an infinite discontinuity.
10. Physics Connection: Instantaneous Velocity
Average velocity is change in position divided by change in time. Average velocity = (s(t2) - s(t1))/(t2 - t1) But physics often asks for velocity at one exact instant. To do that, we let the time interval shrink toward zero. That process is a limit. This is why limits are the gateway to derivatives and why derivatives are so useful in physics.
11. Common Mistakes
- Thinking 0/0 means the answer is automatically 0. - Canceling terms instead of factors. - Forgetting to factor completely. - Forgetting that a limit can exist even when the function is undefined at that point. - Assuming a two-sided limit exists without checking both one-sided limits. - Confusing f(a) with lim x->a f(x).