Question

How do you find the limit `((1/t)-1)/(t^2-2t+1)` as `x->1^+`?

Answer 1

`(1/t-1)/(t^2-2t+1) = (t(1/t-1))/(t(t-1)^2)`

`= (1-t)/(t(t-1)^2)`

`= (-(t-1))/(t(t-1)^2)`

`= (-1)/(t(t-1))`

As `t rarr 1^+` the denominator goes to `0` through positive values and the numerator is negative.

Therefore, the ratio becomes negative and decreases without bound.

`lim_(xrarr1^+)(1/t-1)/(t^2-2t+1) =-oo`