Calculating Limits Using The Limit Laws - Chapter 2.3 Notes

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

2022
Views:

174

Pages:

6
Author:

Nick T.

Calculating Limits Using the Limit Laws Chapter 2.3 Direct Substitution Property: If f is a polynomial or a rational function and a is in the domain of f, then: lim(x→a) f(x) = f(a) Limit Laws: lim(x→a)[f(x) ± g(x)] = lim(x→a)f(x) ± lim(x→a)g(x) lim(x→a)[f(x) * g(x)] = lim(x→a)f(x) * lim(x→a)g(x) lim(x→a)[f(x)/g(x)] = lim(x→a)f(x) / lim(x→a)g(x), g(x) ≠ 0 Exponential Functions: Domain: R Range: (0, ∞) Logarithmic Functions: Domain: (0, ∞) Range: R Trigonometric Functions: sin(x), cos(x): Domain = R tan(x): Domain = R except π/2 ± nπ cot(x): Domain = R except nπ sec(x): Domain = R except π/2 ± nπ csc(x): Domain = R except nπ Indeterminate Forms: 0/0 ∞/∞ ∞ - ∞ 0^∞ ∞^0 1^∞ ∞ * 0 Strategies to Evaluate Indeterminate Forms: Factor and cancel common factors Rationalize the numerator or denominator Use trigonometric identities Multiply by the conjugate L'Hôpital's Rule Exercises: 14. lim(x->-3) (x^2+3x)/(x^2-x-12) = 0/0 (x^2+3x)/(x^2-x-12) = (x(x+3))/((x-4)(x+3)) lim(x->-3) x/(x-4) = -3/(-3-4) = 3/7 16. lim(x->4) (x^2+3x)/(x^2-x-12) = 28/0 19. lim(x->1) (1-x)/(1-x^3) = 0/0 (1-x)/((1-x)(1+x+x^2)) lim(x->1) 1/(1+x+x^2) = 1/3 21. lim(h->0) ((3+h)^2-9)/h = 0/0 (h^2+6h+9-9)/h = 6 22. lim(x->9) (x-9)/(√x-3) = 0/0 (x-9)/(√x-3) * (√x+3)/(√x+3) = (x-9)/(x-9) = 1 23. lim(h->0) ((√(9+h)-3)/h)^2 = 0/0 Rationalize the numerator ((√(9+h)-3)/h)^2 * ((√(9+h)+3))/((√(9+h)+3)) (9+h-9)/(h(√(9+h)+3)) -1/(√(9+0)+3) = -1/6 (h-8)(h-8) h²-8h-8h+64 h²-16h+64 x² = (x+2)(x+2) = (x+2)² A²±B²=(A±B)(A²∓AB+B²) (x+1/h) = (x+1/h) (x+1/h)² = (x+1/h)(x+1/h) (h+1)(h+3) h²+4h+3 lim(h->0) ((3+h)²-9)/h = 0/0 (h²+6h+9-9)/h = 6 lim(x->9) (x-9)/(√x-3) = 0/0 (x-9)/(√x-3) * (√x+3)/(√x+3) = (x-9)/(x-9) = 1 lim(h->0) ((√(9+h)-3)/h)^2 = 0/0 Rationalize the numerator ((√(9+h)-3)/h)^2 * ((√(9+h)+3))/((√(9+h)+3)) (9+h-9)/(h(√(9+h)+3)) -1/(√(9+0)+3) = -1/6 lim(x->2) g(h(x)) = 2 R = 2, L = -2 (h is approaching 2 from below) L ≠ R Limits are not equal to 3 add multiplied 6x² + 45x - 1 T 51-2=49 √99=7 (3-25)/(10+1) = -22/11 = -2 6x²+x-6x-1 x(x+1)-6(x+1) (x-6)(x+1) x=6 x=1 0-15 4(0)-15 / 4(0)+3 = -15/3 = -5 2²-5(2²)+1 8-5(4)+1 = -11 8-20+1 = -11