March 10, 2006 5:25 am
Larson ch 1.3, p. 64 - 66
63.

y = ((1 - Cos[h])^2)/h

Limit[y, h -> 0]

Plot[y, {h, -2 π, 2  π}, 
        PlotStyle -> Blue];

64.

Clear[y]

y = ϕ (Sec[ϕ])

Limit[y, ϕ -> π]

Plot[y, {ϕ, -2 π,  2  π},  
        PlotStyle -> Red];

Plot[y, {ϕ, 2,  4},  PlotStyle -> Red];

65.

Clear[y]

y = Cos[x]/Cot[x]

Limit[y, x -> π/2]

Plot[y, {x, -2 π, 2  π}, 
        PlotStyle  -> Red];

66.

Clear[y]

y = (1 - Tan[x])/(Sin[x] - 
            Cos[x])

Limit[y, x -> π/4]

Plot[y, {x, -2 π, 2  π}, 
        PlotStyle -> Red];

Plot[y, {x, -2 π, π/4}, 
        PlotStyle -> Blue];

Plot[y, {x, -1, π/4}, PlotStyle -> Blue];

67.

Clear[y]

y = ((Sin[t])^2)/t^2

Limit[y, t -> 0]

Plot[y, {t, -2 π, 2  π},  
        PlotStyle -> Blue];

85.

Clear[y]

f = x

g = Sin[x]

h = Sin[x]/x

p1 = 
        Plot[f, {x, -2 π, 2  π}, 
          PlotStyle -> Green];

p2 = 
        Plot[g, {x, -2 π, 2  π}, 
          PlotStyle -> Red];

p3 = 
        Plot[h, {x, -2 π, 2  π}, 
          PlotStyle -> Blue];

Show[p1, p2, p3];

86.

f2 = x

g2 = (Sin[x])^2

h2 = ((Sin[x])^2)/x

p4 = 
        Plot[f2, {x, -2 π, 2  π}, 
          PlotStyle -> Green];

p5 = 
        Plot[g2, {x, -2 π, 2  π}, 
          PlotStyle -> Red];

p6 = 
        Plot[h2, {x, -2 π, 2  π}, 
          PlotStyle -> Blue];

Show[p4, p5, p6];

105.

Clear[f]

f = (Sec[x] - 1)/x^2

Plot[f, {x, -2 π, 2  π}, 
        PlotStyle -> Red];

Plot[f, {x, - .2,  .2}, PlotStyle -> Red];

Limit[f, x -> 0]

Limit[Cos[x], x -> 0]