Guide to Limits in Calculus
Comprehensive guide to Limits in Calculus, designed for students in the MCV4U course.
1. Introduction to Limits
A limit describes the value that a function f(x) approaches as the input x approaches a particular value. Limits are crucial for understanding continuity, derivatives, and integrals in calculus.
Notation:
\[ \lim_{x \to c} f(x) = L \]
This means as \(x\) approaches \(c\), \(f(x)\) approaches \(L\).
2. Types of Limits
a) Finite Limits at Finite Points
If \(f(x)\) approaches a specific value \(L\) as \(x \to c\), the limit exists.
\[ \lim_{x \to 2} (3x + 1) = 7 \]
Solution: Substitute \(x = 2\): \[ f(2) = 3(2) + 1 = 7 \]
b) Limits at Infinity
Describes the behavior of \(f(x)\) as \(x \to \infty\) or \(x \to -\infty\).
\[ \lim_{x \to \infty} \frac{1}{x} = 0 \]
As \(x\) becomes infinitely large, \( \frac{1}{x} \) approaches 0.
c) Infinite Limits
Occurs when \(f(x) \to \infty\) or \(f(x) \to -\infty\) as \(x \to c\).
\[ \lim_{x \to 0^+} \frac{1}{x} = \infty \]
As \(x\) approaches 0 from the right, \(\frac{1}{x}\) grows without bound.
3. One-Sided Limits
One-sided limits consider the behavior of \(f(x)\) from only one side of \(x = c\).
- Right-hand limit: \(\lim_{x \to c^+} f(x)\)
- Left-hand limit: \(\lim_{x \to c^-} f(x)\)
Evaluate \(\lim_{x \to 2} \frac{|x - 2|}{x - 2}\):
- For \(x \to 2^+\): \(\frac{|x - 2|}{x - 2} = 1\)
- For \(x \to 2^-\): \(\frac{|x - 2|}{x - 2} = -1\)
Since the one-sided limits are not equal, the limit does not exist.
4. Special Techniques for Evaluating Limits
a) Direct Substitution
Substitute \(x = c\) directly into \(f(x)\).
\[ \lim_{x \to 3} (x^2 + 2x + 1) = 3^2 + 2(3) + 1 = 16 \]
b) Factoring
Useful when substitution results in an indeterminate form like \(0/0\).
\[ \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \]
Factor the numerator: \((x - 2)(x + 2)\). Cancel \(x - 2\): \[ \lim_{x \to 2} (x + 2) = 4 \]
c) Rationalizing
Useful for square roots.
\[ \lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x} \]
Multiply by the conjugate, simplify, and find the limit: \[ \frac{1}{2} \]
5. Limits and Asymptotes
Vertical Asymptotes: Occur when \(f(x) \to \infty\) or \(-\infty\) as \(x \to c\).
Horizontal Asymptotes: Describe \(f(x)\) as \(x \to \infty\).
6. Graphing Limits
Use limits to determine asymptotes, intercepts, and behavior near critical points for accurate graphing.
Summary
By understanding limits, one can solve problems involving continuity, asymptotes, and behavior at infinity with confidence.