Properties
of
Limits
, Properties of Limits
The Definition
In the preceding articles we defined tangent slopes to be limits of secant
slopes, and instantaneous rates of change (velocities, marginal cost) to be
limiting values of average rates of change (average velocity, average cost).
We found these limits from such
expressions as
Average change in f over the fxAx) -flx)
interval from x to x + Ax Ax (1)
In each able to change the expression on the right into one inn
case we were
which we could set Ax equal to 0 to calculate the limit we
wanted.
The problem is that we cannot always change the
right-hand side of
Eq. (1) into a form in which we can substitute 0 for Ax. For
example.
suppose we wanted to find the slope of the curve y =sin x by
letting x
approach 0 in the expression
sin (x + Ax) - sin x
Ax 2)
1
of
Limits
, Properties of Limits
The Definition
In the preceding articles we defined tangent slopes to be limits of secant
slopes, and instantaneous rates of change (velocities, marginal cost) to be
limiting values of average rates of change (average velocity, average cost).
We found these limits from such
expressions as
Average change in f over the fxAx) -flx)
interval from x to x + Ax Ax (1)
In each able to change the expression on the right into one inn
case we were
which we could set Ax equal to 0 to calculate the limit we
wanted.
The problem is that we cannot always change the
right-hand side of
Eq. (1) into a form in which we can substitute 0 for Ax. For
example.
suppose we wanted to find the slope of the curve y =sin x by
letting x
approach 0 in the expression
sin (x + Ax) - sin x
Ax 2)
1
