Using the limit definition, how do you differentiate #f(x) = x^2 - 1598#?
1 Answer
Expand and simplify. Then evaluate the limit.
Explanation:
For this function, we get
If we try to evaluate the limit by substitution, we get the indeterminate form
It may not be clear that expanding the numerator will help, but we need to try something, so let's do it and see what happens.
# = lim_(hrarr0)(color(red)(x^2)+2xh+h^2color(blue)(-1598) color(red)(-x^2)color(blue)(+1598))/h#
# = lim_(hrarr0)(2xh+h^2)/h#
If we try substitution, we still get indeterminate form
Note however, that for all
Therefore,
In many classes it is permissible to write this without comment as follows:
#= lim_(hrarr0)([x^2+2xh+h^2-1598] - [x^2-1598])/h#
# = lim_(hrarr0)(x^2+2xh+h^2-1598 -x^2+1598)/h#
# = lim_(hrarr0)(2xh+h^2)/h#
# = lim_(hrarr0)(2x+h)#
# = 2x#