vertical asymptotes of (limit of |x-3|/(|5-x|-|1-x|) x->3-)

Question

vertical asymptotes of (limit of |x-3|/(|5-x|-|1-x|) x->3-)

As the title said I'm not sure whether this equation have vertical asymptotes or not

http://www5a.wolframalpha.com/Calculate/MSP/MSP26522d5i9h0h3afgi6100005dbd6gc0ehhf3cdg?MSPStoreType=image/gif&s=62&w=290.&h=112.&cdf=Coordinates&cdf=Tooltips

Answer 1

We are interested in the behaviour of our function when $x$ is near $3$ but a little below.

For such $x$, we have $|5-x|=5-x$ and $|1-x|=x-1$. The difference is $6-2x$, that is, $2(3-x)$.

Thus for $x$ near $3$ but below $3$, the ratio is exactly $\frac{1}{2}$. No blowing up. No vertical asymptote.

Note that on the other side of $3$ but close to $3$, the ratio is $-\frac{1}{2}$.