3x2- 11x -4 1. (6 pts) Calculate lim *+-00 12 + x - x2 If no finite limit exists, be as precise as possible among co, -co, or DNE. Justify your answer. Tables alone will not suffice as justification.Let c be a constant and let f and g be functions such that the limits lim f(x) Land lim g(a) M exist. We have the following algebraic rules for limits: 1. Sum Law lim If (a) + 9(x)] = lim f (x] + lim q (x) = L+M x7a * >a 2. Difference Law lim If(x) - 9(x)] = lim f (x) - lim q (x) = L-M x79 3. Constant Multiple Law lim lef (x)] = c lim f(x) = c L xza 4. Product Law lim [f(:) . g(2)] = lim f(x] . lim q (x) = LM x7a X 7a 5. Quotient Law If lim g() * 0, then lim f(:) lim f(x ) L g(I) x 7a 3 lim 9 ( x) 6. Power Law If n is a positive integer, then lim [f(*)]" = (lim f(x))"= Ln x 70Let c be a constant and let f and g be functions such that the limit lim f(:) exists. We have the following algebraic rules for limits: 7. Constant Law lim c = C - y=c 8. Identity Law lima = a 9 n 9 4 , 4 = X 9. If n is a positive integer, then lim &" = ( lim x)= a x-70 10. If n is a positive integer, then lim VI (lim x ) xin [If n is even, we assume a > 0. xza a 11. If n is a positive integer, then lim ) X 7a If n is even, we assume lim f(x) >0.]Technique 3: Re-writing the Function Consider the function f(x) = |x -5). Rewrite f as a piecewise function. 1x-51 My = - (x- 5) f (x )= 1x-51= _x-5 x25 - (x-5), x45 y= x - 5 7X 5 Example 6 Evaluate lim 6 - 2c = >3 12 - 3cl when we plug in x=3 we get 0/0. so we must investigate y= 1 x2- 3x/ = 1 (x(x-3)) further Note 1y gy = x (x -3 ) y= 1x(x-3)1 1x - 3x/ = x 23x XSO > X - ( x 2 - 3x ) , OCX a a The Squeeze Theorem If f(x) 0, then lim O I FOOT if ris then x exz is not defined If r is a rational number, r > 0, such that a" is defined for all a, then for negative values of x limIntuitive Definition of Limits at Infinity Let f be a function defined on some interval (a, co). Then lim f(x) = L means that the values of f can be made arbitrarily close to _ by requiring a to be sufficiently large. Let f be a function defined on some interval (co, a). Then lim f(a) = L means that the values of f can be made arbitrarily close to L by requiring a to be sufficiently large and negative.Theorem Let f(@) = a(b), where a, b are real numbers, b > 1. Then lim a(b)* = 0 I ) 00 a zo lim a(b)= = - OG aco