how to find vertical asymptotes

To find the vertical asymptotes of a function, especially a rational function (a fraction of polynomials), follow these steps:

1. Factor the numerator and denominator of the function, if possible.
2. Simplify the function by canceling any common factors that appear in both numerator and denominator. These canceled factors correspond to holes, not vertical asymptotes.
3. Set the denominator equal to zero and solve for the variable (usually xxx). The solutions give the xxx-values where vertical asymptotes occur.
4. Exclude any values that were canceled out during simplification, as those correspond to holes, not asymptotes.

The vertical asymptotes are vertical lines x=ax=ax=a where the function tends to infinity or negative infinity as xxx approaches aaa. Example: For the function g(x)=x−2x2−4x+3g(x)=\frac{x-2}{x^2-4x+3}g(x)=x2−4x+3x−2​:

• Factor the denominator: x2−4x+3=(x−3)(x−1)x^2-4x+3=(x-3)(x-1)x2−4x+3=(x−3)(x−1).
• The function is x−2(x−3)(x−1)\frac{x-2}{(x-3)(x-1)}(x−3)(x−1)x−2​.
• There are no common factors to cancel.
• Set denominator equal to zero: x−3=0x-3=0x−3=0 or x−1=0x-1=0x−1=0, so x=3x=3x=3 or x=1x=1x=1.
• Vertical asymptotes are at x=3x=3x=3 and x=1x=1x=1

This method applies broadly to rational functions. For other types of functions:

• Logarithmic functions f(x)=log⁡(ax+b)f(x)=\log(ax+b)f(x)=log(ax+b) have vertical asymptotes where ax+b=0ax+b=0ax+b=0.
• Trigonometric functions like tan⁡x\tan xtanx, sec⁡x\sec xsecx, csc⁡x\csc xcscx, and cot⁡x\cot xcotx have vertical asymptotes at specific periodic points.
• Polynomial and exponential functions generally do not have vertical asymptotes

In summary, the key step is to find values that make the denominator zero after simplification, and these values correspond to vertical asymptotes unless canceled out