Limits
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. What is a Limit?
Definition:
A limit describes the value that a function approaches as the input approaches a certain point
\[ \lim_{x \to a} f(x) = L \]
Read as: "The limit of f(x) as x approaches a equals L"
• x approaches a but never necessarily equals a
• The function doesn't need to be defined at x = a
• We care about values near a, not at a
2. One-Sided Limits
Left-Hand Limit:
Approaches from the left (values less than a)
\[ \lim_{x \to a^-} f(x) = L \]
Right-Hand Limit:
Approaches from the right (values greater than a)
\[ \lim_{x \to a^+} f(x) = L \]
Key Rule:
A two-sided limit exists if and only if both one-sided limits exist and are equal:
\[ \lim_{x \to a} f(x) = L \quad \text{if and only if} \quad \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \]
3. When Does a Limit Exist?
Limit Exists When:
✓ Left-hand limit = Right-hand limit
✓ Both one-sided limits are finite numbers
✓ Function approaches the same value from both sides
Limit Does NOT Exist When:
✗ Left-hand limit ≠ Right-hand limit
✗ Function approaches ∞ or -∞
✗ Function oscillates without settling to a value
4. Limits at Vertical Asymptotes
Definition:
At a vertical asymptote x = a, the function approaches infinity or negative infinity
Possible Cases:
• \( \lim_{x \to a^-} f(x) = +\infty \) and \( \lim_{x \to a^+} f(x) = +\infty \)
• \( \lim_{x \to a^-} f(x) = -\infty \) and \( \lim_{x \to a^+} f(x) = -\infty \)
• \( \lim_{x \to a^-} f(x) = +\infty \) and \( \lim_{x \to a^+} f(x) = -\infty \) (or vice versa)
Example:
For \( f(x) = \frac{1}{x} \) at x = 0:
\( \lim_{x \to 0^-} \frac{1}{x} = -\infty \)
\( \lim_{x \to 0^+} \frac{1}{x} = +\infty \)
x = 0 is a vertical asymptote
5. End Behavior (Limits at Infinity)
Definition:
End behavior describes what happens to f(x) as x approaches positive or negative infinity
\[ \lim_{x \to \infty} f(x) \quad \text{and} \quad \lim_{x \to -\infty} f(x) \]
Key Rules:
• \( \lim_{x \to \infty} \frac{c}{x^n} = 0 \) for any positive integer n
• Horizontal asymptote at y = L if limit equals L
• For rational functions, compare degrees of numerator and denominator
6. Limit Laws
Basic Laws:
If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then:
1. Sum Law:
\[ \lim_{x \to a} [f(x) + g(x)] = L + M \]
2. Difference Law:
\[ \lim_{x \to a} [f(x) - g(x)] = L - M \]
3. Constant Multiple Law:
\[ \lim_{x \to a} [c \cdot f(x)] = c \cdot L \]
4. Product Law:
\[ \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M \]
7. Division, Power, and Root Laws
Quotient Law:
\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M} \quad \text{(if } M \neq 0 \text{)} \]
Power Law:
\[ \lim_{x \to a} [f(x)]^n = L^n \]
Root Law:
\[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L} \]
(For all L if n is odd; for L ≥ 0 if n is even)
8. Limits of Polynomials and Rational Functions
For Polynomials:
Simply substitute x = a:
\[ \lim_{x \to a} p(x) = p(a) \]
Example:
\( \lim_{x \to 2} (x^2 + 3x - 1) = 2^2 + 3(2) - 1 = 9 \)
For Rational Functions:
If denominator ≠ 0 at x = a, substitute directly:
\[ \lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)} \quad \text{(if } q(a) \neq 0 \text{)} \]
⚠️ If you get \( \frac{0}{0} \) (indeterminate form), use factorization or rationalization!
9. Limits by Factorization
Method:
When substitution gives \( \frac{0}{0} \):
1. Factor numerator and denominator
2. Cancel common factors
3. Substitute x = a into simplified expression
Example:
Find: \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \)
Direct substitution: \( \frac{0}{0} \) (indeterminate)
Factor: \( \frac{(x-2)(x+2)}{x-2} \)
Cancel: \( x + 2 \) (for x ≠ 2)
Substitute: \( 2 + 2 = 4 \)
Answer: 4
10. Limits by Rationalization
Method:
When expression involves square roots and gives \( \frac{0}{0} \):
1. Multiply by conjugate of numerator or denominator
2. Use difference of squares: \( (a-b)(a+b) = a^2 - b^2 \)
3. Simplify and cancel common factors
4. Substitute x = a
Example:
Find: \( \lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x} \)
Multiply by conjugate: \( \frac{\sqrt{x+1} - 1}{x} \cdot \frac{\sqrt{x+1} + 1}{\sqrt{x+1} + 1} \)
= \( \frac{(x+1) - 1}{x(\sqrt{x+1} + 1)} = \frac{x}{x(\sqrt{x+1} + 1)} \)
Cancel x: \( \frac{1}{\sqrt{x+1} + 1} \)
Substitute x = 0: \( \frac{1}{\sqrt{1} + 1} = \frac{1}{2} \)
Answer: \( \frac{1}{2} \)
11. Important Special Limits
1. Constant Function:
\[ \lim_{x \to a} c = c \]
2. Identity Function:
\[ \lim_{x \to a} x = a \]
3. Limit of Reciprocals:
\[ \lim_{x \to 0^+} \frac{1}{x} = +\infty, \quad \lim_{x \to 0^-} \frac{1}{x} = -\infty \]
12. Quick Reference Summary
Key Strategies:
1. Try direct substitution first
2. If you get \( \frac{0}{0} \), factor or rationalize
3. If you get \( \frac{k}{0} \) (k ≠ 0), limit is ±∞
4. Check one-sided limits if limit doesn't exist
5. For limits at ∞, compare degrees of polynomials
📚 Study Tips
✓ Limit exists when left and right limits are equal and finite
✓ Always try direct substitution first before other methods
✓ Indeterminate form \( \frac{0}{0} \) requires algebraic simplification
✓ Use conjugates when dealing with square roots
✓ Limit laws only work when individual limits exist