The numerator goes to 1, and the denominator goes to zero. Since the denominator is always positive, the limit is positive infinity. Done.
"plugging in values" doesn't evaluate the limit, in some unusual cases, a function can fluctuate quite wildly near the point the limit is evaluated at.
in your case, what you would need to show that no matter how big a number N you choose, you can pick a value of t close enough to 1 so that
t^3/(t^2 -1)^2 > N. the value that will work, will depend on the number N.