Vertical asymptotes could be find in the points where the Exist Conditions of f(x) are not satisfied
The Field of Existence of f(x) is:
x in ]-oo,3[uu]3,7[uu]7,+oo[
because
f(x)=(N(x))/(D(x))
FE (Field of Existence):
D(x)!=0
(x-3)(x-7)!=0
x_1!=3
x_2!=7
Now these values of x are vertical asymptotes if:
lim_(x rarr x_(1,2)^+-)f(x)=+-oo
lim_(x rarr 3^-)f(x)=lim_(x rarr 3^-)1/((2.9-3)(2.9-7))=
lim_(x rarr 3^-)1/((0^-)(-4.1))=lim_(x rarr 3^-)1/0^+=+oo
lim_(x rarr 3^+)f(x)=lim_(x rarr 3^+)1/((3.1-3)(3.1-7))=
lim_(x rarr 3^+)1/((0^+)(-3.9))=lim_(x rarr 3^+) 1/(0^-) =-oo
lim_(x rarr 7^-)f(x)=lim_(x rarr 7^-)1/((6.9-3)(6.9-7))=
lim_(x rarr 7^-)1/((3.9)(0^-))=lim_(x rarr 7^-)1/(0^-)=-oo
lim_(x rarr 7^+)f(x)=lim_(x rarr 7^+)1/((7.1-3)(7.1-7))=
lim_(x rarr 7^+)1/((4.1)(0^+))=lim_(x rarr 7^+)1/(0^+)=+oo