Question

1. Find lim f(x) and lim f(x) for each of the following. Assign oo or - where appropriate: (a) f(z)=42 -2 (b) f(x)= 3r-2 (e) f(1)--5rt6 (x - 2)2 2. Find each of the f limits, assigning oo or-o where appropriate x+2z-1 (b)lm 3r-2 3 r-2 (c) li 42+1 (k) im (2r V4r2-8r 3) 15r-2 3. Find the horizontal and vertical asymptotes of each of the following functions 4. Sketch the graph of each of the following functions and determine, with reasons, using the defini tion of continuity, whether each is continuous at the given value (c) a)-2-r if x <3

Answer 1

f(x) = 4 - x^3/x - 2 lim_x rightarrow 2^- f(x) = infinity lim_x rightarrow 2^+ f(x) = -infinity f(x) = 3x - 2/x^2 - 6x + 8 = 3x - 2/(x - 4) (x - 2) lim_x rightarrow 2^- f(x) = infinity lim_x rightarrow 2^+ f(x) = infinity f(x) = x^2 - 5x + 6/(x - 2)^2 = (x - 3) (x - 2)/(x - 2)^2 = x - 3/x - 2 lim_x rightarrow 2^- f(x) = infinity lim_x rightarrow 2^+ f(x) = -infinity \r\n lim_x rightarrow infinity 4x^2 + 2x - 1/8x^2 - 3x + 7 = lim_x rightarrow infinity x^2 (4 + 2/x - 1/x^2)/x^2 (8 - 3/x + 7/x^2) we know lim_x rightarrow infinity 1/x = 0 lim_x rightarrow infinity 4 + 2/x - 1/x^2/8 - 3/x + 7/x^2 = 4/8 = 1/2 lim_x rightarrow -infinity 4x^2 + 1/(x + 1)^3 = lim_x rightarrow -infinity x^2 (4 + 1/x^2)/[x (1 + 1/x)]^3 = lim_x rightarrow -infinity 4 + 1/x^2/x (1 + 1/x)^3 = 4/-infinity = 0 \r\n lim_x rightarrow infinity (x^2/3x - 2 - x/3) = lim_x rightarrow infinity