Introduction to limits, derivatives, differentiation rules, and optimization problems. Build the foundation for understanding rates of change and analyzing functions.
Calculate the limit!
Function: f(x) = x²
Find the derivative!
Function: f(x) = x²
The limit of f(x) as x approaches a is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a (but may not equal a).
Notation: lim f(x) as x → a = L
Problem: Find lim x² as x → 3
Step 1: Since x² is continuous, lim x² as x → 3 = f(3)
Step 2: f(3) = 3² = 9
Answer: lim x² as x → 3 = 9
Problem: Find lim (x² - 4)/(x - 2) as x → 2
Step 1: Factor numerator: (x² - 4) = (x - 2)(x + 2)
Step 2: Cancel: (x - 2)(x + 2)/(x - 2) = x + 2 (for x ≠ 2)
Step 3: lim (x + 2) as x → 2 = 2 + 2 = 4
Answer: lim = 4