How do you find the limit of #(5x^2-8x-13)/(x^2-5)# as x approaches 3?

1 Answer
Mar 6, 2015

Use the properties of limits.

#lim_(xrarr3)(x^2-5)=lim_(xrarr3)x^2-lim_(xrarr3)5# (Subtraction property)
#=(lim_(xrarr3)x)^2-lim_(xrarr3)5# (Power property)
#=3^2-5# (Limits of Identity and constant functions)
#=9-5=4# (Arithmetic)

Because the limit of the denominator is not #0#, we can use the quotient property for limits.

By steps similar to those above, we can show that
#lim_(xrarr3)(5x^2-8x-13)=5(3)^2-8(3)-13=45-37=8#.

And, therefore #lim_(xrarr3)(5x^2-8x-13)/(x^2-5)=8/4=2#.