Continuity/ Limits involving infinity/ Starting Differentiation

1. Evaluate the following limits: 6pt (a) lim 4 X -+2 (b) lim 2x x-+2 (c) x -3 lim x-+2 x+1 (d) lim x2 - 4 x -12 x - 2 (e) lim Vtz + 9 -3 t2 (f ) lim X - 7x + 10 x - 2 (g) lim Vxth- vx h-+ h 2. Evaluate the following limits involving infinity: 6pt (a) lim 1 - x2 x-14- x2 - 16 (b) 1-x3 lim x-+00 2x2 - 7 (c) lim 1-x-x2 x-100 2x2 - 7 (d) lim [x* + x5] (e) lim Vx2 - 4 -+ 0o x +1 (f) lim vx2 + ax - Vx2 + bx 3. Assume that constant a is positive. Find equations for all horizontal and vertical asymptotes for the graph of f(x) _ Vax' + 4 x - b 3pt 4. Find the formula for a single function f(x) that satisfies all the following conditions: 3pt (a) limx-+too f(x) = 0 (b) limx-to f(x) = -oo (c) limx-3- f(x) = 0o (d) limx-,3+ f(x) = -0o (e) f(2) = 0 This one is a toughy :) 5. Find the derivative of the function f(x) = 3x2 - x by using the definition of the derivative and use the derivative to find the equation of the tangent line at the point P(-1,4) 3pt 6. Find the slant asymptote of f(x) = 2pt 7. Differentiate: 4pt (a) f(x) = (2x3+ 3) (x4 - 2x) (b) f ( x) = vx3 (c) f (r ) = 1-V (d) f( x) = x Where c is a constant. You do not need to simplify any solutions. 8. Differentiate: 4pt (a) y = tan'(sec- (x*) ) (b) y = sec(0)tan(0) (c) y = 1+sec(@) (d) y = cot?(sin(0))