having lots of trouble finding limits

1:

lim as v -> -2 from left of {v-2 / |v-2|}

2:

lim as v -> -2 from right of {v-2 / |v-2|}

for these 2, i'm thinking that |v-2| is 2 - v if v <= 2 and v - 2 if x > 2

i got that far but don't know what to do next

3:

lim as x -> 1 of cos^-1 {1 - sqrt(x) / 1 - 2}

the cos^-1 is what has me stuck, not sure how to interpret

4:

lim as x-> infinity of {sqrt(x^2 -9) / 2x - 6}

what i did was multiply top/bottom by sqrt(x^2 -9) to make numerator x^2 - 9 but then the not sure what to do w/ the denominator

5:

lim as x -> infinity of {x^4 + 2 / (x^2 - 1) (2x^2 + 1)

this one i tried to factor out the numerator but it doesn't factor out properly for it to cancel out the denominator's (x^2 - 1). unless i factored wrong, what am i missing/doing wrong with this one?