Is the graph of #f (x) = (X^2 + x)/x# continuous on the interval [-4, 4]?

1 Answer
Feb 9, 2017

Answer:

see below

Explanation:

To show that the function #f(x)=(x^2+x)/x# is continuous on [-4,4] we need to do the following:

  1. Show that #lim_(x->-4^+) (x^2+x)/x =f(-4)#

  2. Show that #lim_(x->4^-) (x^2+x)/x =f(4)#

That is,

  1. #lim_(x->-4^+) (x^2+x)/x =lim_(x->-4^+) (x(x+1))/x#

# =lim_(x->-4^+)(cancelx(x+1))/cancelx=lim_(x->-4^+) x+1 = -3#

#f(-4)=(16-4)/-4 =12/-4 = -3#

Since #lim_(x->-4^+) (x^2+x)/x =f(-4)=-3#, f is continuous from the right at x = - 4.

2.# lim_(x->4^-) (x^2+x)/x = lim_(x->4^-) (x(x+1))/x #

# =lim_(x->4^-)(cancelx(x+1))/cancelx=lim_(x->4^-) x+1 = 5#

#f(4)=(16+4)/4=20/4=5#

Since #lim_(x->4^-) (x^2+x)/x =f(4)#, f is continuous from the left at 4.

Since f is continuous from the left at x = 4and continuous from the right at x = -4, hence f is continuous on the interval [-4,4].