How do you compute the rate of change of the function f(x) = 1+1/(x+1) at the point x = a?

1 Answer
Jim H
Aug 3, 2017

Using the definition, see below.

Explanation:

The rate of change at x=a can be found using

lim_(hrarr0)(f(a+h) - f(a))/h

or

lim_(xrarra)(f(x) - f(a))/(x-a).

Using the second form, we have

lim_(xrarra)(f(x) - f(a))/(x-a) = lim_(xrarra)((1+1/(x+1)) - (1+1/(a+1)))/(x-a)

= lim_(xrarra)(1/(x+1) - 1/(a+1))/(x-a)

= lim_(xrarra)(((a+1)-(x+1))/((x+1)(a+1)))/((x-a)/1)

= lim_(xrarra)(a-x)/((x+1)(a+1)) * 1/(x-a)

= lim_(xrarra)(-1(cancel(x-a)))/((x+1)(a+1)) * 1/(1(cancel(x-a))

= (-1)/((a+1)(a+1)) = (-1)/(a+1)^2

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