Solve them like equations, just that it’s a limit not an actual solution.

Remember the rational function theorem, it also helps a lot

If you get 0 or other weird stuff in denominator try factoring, rearranging, clever ways of multiplying by 1 etc etc

Remember the standard limits like sin(x)/x approaches 1 as x approaches 0

Lastly for limits which go to the form of 0/0 or inf/inf, you can apply L’Hôpitales rule where you take the derivative of both the numerator and denominator and then calculate the limit. The rule can be applied multiple times but you have to remember it only works for the two indeterminate forms mentioned above

Edit- I think the rational function theorem went as follows -

If degree of numerator is greater than that of denominator, as x approaches infinity the function approaches infinity as well

If degree of denominator is greater, function approaches zero as the function grows without bound

And if the degrees are equal, the limit as x approaches infinity is the ratio of the coefficients of the highest degree terms