Precalculus Notes: Limits
Precalculus: Limits - A Preview of Calculus
1. Introduction to Limits
- Goal: To understand what happens to the values of a function f(x) as the variable x approaches a specific number a.
2. Definition of a Limit
- If we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a, but not equal to a, then we write:
\lim_{x \to a} f(x) = L - This is read as “the limit of f(x), as x approaches a, equals L.”
- Alternative notation: f(x) \to L as x \to a. This is read as “f(x) approaches L as x approaches a.”
3. Graphical Interpretation of Limits
- The limit of a function as x approaches a describes the behavior of the function near x = a, not necessarily at x = a.
- The value of f(a) itself is irrelevant to the existence or value of the limit.
- Cases:
- f(a) may not be defined.
- f(a) may be defined, but f(a) ≠ L.
- Regardless, if \lim_{x \to a} f(x) = L , the limit exists and is equal to L.
4. Estimating Limits
- Methods:
- Numerically: Using tables of values to observe the behavior of f(x) as x gets closer to a.
- Graphically: Using graphs to visualize the behavior of f(x) as x approaches a.
- These methods are used to estimate limits. Exact values are determined using techniques to be developed in Section 12-2.
4.1. Example 1: Estimating a Limit Numerically and Graphically
- Problem: Guess the value of \lim_{x \to 2} \frac{x^2 + x - 6}{x - 2} .
- Numerical Approach: Examine values of the function as x approaches 2.
- Graphical Approach: Analyze the graph of the function near x = 2.
4.2. Example 2: Finding a Limit from a Table
- Create a table of values for a function as x approaches a certain value (e.g., 0) from both sides.
- Observe the corresponding y values to infer the limit.
5. Limits That Fail to Exist
- Functions do not necessarily approach a finite value at every point.
- It is possible for a limit not to exist.
5.1. Example 3: Function with a Jump
- If the limit from the left and the limit from the right are not equal, then the limit does not exist (DNE).
- \lim{x \to a^-} f(x) ≠ \lim{x \to a^+} f(x) \implies \lim_{x \to a} f(x) \text{ does not exist}
5.2. Example 4: Function that Oscillates
- If the function oscillates too wildly as x approaches a, the limit does not exist.
5.3. Example 5: Function with a Vertical Asymptote
- If the function approaches infinity (or negative infinity) as x approaches a, the limit does not exist.
- Example: \lim_{x \to 2} \frac{1}{x-2} = DNE
6. One-Sided Limits
- Right-Hand Limit: \lim_{x \to a^+} f(x) = L means that as x approaches a from the right, f(x) approaches L.
- Left-Hand Limit: \lim_{x \to a^-} f(x) = L means that as x approaches a from the left, f(x) approaches L.
7. Two-Sided Limits vs. One-Sided Limits
- The two-sided limit exists if and only if both one-sided limits exist and are equal:
\lim{x \to a} f(x) = L \iff \lim{x \to a^-} f(x) = L \text{ and } \lim_{x \to a^+} f(x) = L
8. Homework