Question 295503
Simplify first. 
You can factor numerator and denominator and get rid of common factors, if there are any.
{{{x^2+5x-6=(x+6)(x-1)}}}
{{{x^2+2x-3=(x+3)(x-1)}}}
{{{(x^2+5x-6)/(x^2+2x-3)=((x+6)cross(x-1))/((x+3)cross(x-1))}}}
{{{(x^2+5x-6)/(x^2+2x-3)=(x+6)/(x+3))}}}
Vertical asymptotes occur when the denominator goes to zero. 
For f(x), the denominator goes to zero when,
{{{x=-3}}}
To find horizontal asymptotes, divide numerator and denominator by the highest exponent x term, then take the limit.
{{{f(x)=(x^2+5x-6)/(x^2+2x-3)}}}
{{{f(x)=(x^2/x^2+5x/x^2-6/x^2)/(x^2/x^2+2x/x^2-3/x^2)}}}
{{{f(x)=(1+5/x-6/x^2)/(1+2/x-3/x^2)}}}
in the limit, the crossed out terms go to zero.
{{{f(x)=(1+cross(5/x)-cross(6/x^2))/(1+cross(2/x)-cross(3/x^2))}}}
So then
{{{lim(x->infinity, f(x))=1}}}
There is a horizontal asymptote at {{{y=1}}}.
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Since there is a division by zero at {{{x=-3}}}, the value of {{{f(-3)}}} is undefined. 
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 {{{drawing( 300, 300, -10, 10, -5, 5,green(line(-3,-10,-3,10)),blue(line(-20,1,20,1)),
 graph( 300, 300, -10, 10, -5, 5, (x^2+5x-6)/(x^2+2x-3))) }}}