University:
York University
Course:
MATH 1013 | Applied Calculus I
Academic year:

2022
Views:

490

Pages:

4
Author:

Nick T.

Chapter 2 Limits and Derivatives lim [f(x) + g(x)] = lim f(x) + lim g(x) x→a x→a x→a lim [f(x) - g(x)] = lim f(x) - lim g(x) x→a x→a x→a lim [f(x) * g(x)] = lim f(x) * lim g(x) x→a x→a x→a lim [f(x) / g(x)] = lim f(x) / lim g(x) x→a x→a x→a g(x) ≠ 0 lim [f(x)]^n = [lim f(x)]^n n is an integer x→a x→a lim c = c x→a lim x = a x→a lim x^n = a^n x→a n is a positive integer lim √x^n = √a^n x→a n is even, a > 0 Root law: lim √f(x) = √lim f(x) x→a x→a Continuity A function f is continuous at a number a if: f(a) is defined (that is, a is in the domain of f) lim f(x) exists x→a lim f(x) = f(a) x→a Continuity graph with various functions, including a continuous function, a function with a removable discontinuity, a function with a jump discontinuity, and a function with an infinite discontinuity Infinite Discontinuities = Asymptotes cos 0 = 1 sin 0 = 0 Cosine and Sine functions are continuous at 0 tan x = sin x / cos x Discontinuous at: x = ±π/2, ±3π/2, ±5π/2, ... Period = π Infinite Limits lim f(x) = ±∞ x→a Horizontal and Vertical Asymptotes Trigonometry Identities: cos²θ + sin²θ = 1 1 - sin²θ = cos²θ cos²θ - sin²θ = cos2θ (1 + √2sinθ)(1 - √2sinθ) = 1 - 2sin²θ 2.6 Limits at Infinity and Infinite Limits Limits at Infinity: lim f(x) = L x→∞ Infinite Limits: lim f(x) = ∞ x→a To find horizontal asymptotes, calculate: lim f(x) = L x→∞ To find vertical asymptotes: Find zeros of the denominator of a rational function. Example: lim (x^2 - 4) / (x - 2) x→2 f(x) = (x^2 - 4) / (x - 2) The vertical asymptote is x = 2. f(x) has a horizontal asymptote y = n