March 24, 2006 7:10 am
Larson ch 1.4, p. 75 - 78.
14.

y = (Sqrt[x] - 2)/(x - 4)

Limit[y, x -> 4, Direction -> -1]

p1 = Plot[y, {x, 0, 5}, PlotStyle -> Blue];

Clear[y, p1]

15.

y = Abs[x]/x

Limit[y, x -> 0]

Plot[y, {x, -5, 5}, PlotStyle -> Red];

Interesting. The limit fails to exist since the right and left limits do not agree. Yet Mathematica gives an answer of 1. This at least illustrates the value of obtaining a plot as well as an analytic solution.
16.

Clear[y, p1]

y = Abs[x - 2]/(x - 2)

Limit[y, x -> 2]

Limit[y, x -> 2, Direction -> 0]

Super! Setting the Direction to zero is equivalent to computing both the right and left sided limits. Let me try a quick check.

Clear[y]

y = x

Limit[y, x -> 0, Direction -> 0]

Perfect.
17.

Clear[y]

y = ((1/(x + dx)) - 1/x)/dx

Limit[y, dx -> 0, Direction -> 1]