How to Solve Limits at Infinity Using Algebra
Even when you can find a limit using a calculator, it’s better to solve the problem algebraically because then you have a mathematically airtight answer. The calculator answer may be very convincing, but it’s not mathematically rigorous, so if you stop there, the math police may get you.
Here’s an example: try some algebra for the problem

Try substitution — always a good idea.
No good. You get
which tells you nothing. On to plan B.
Remember that
does not equal zero. Substitution does not work for this problem. If you plug
into x, you get
which does not necessarily equal zero
As a result,
tells you nothing about the answer to a limit problem.
Because
contains a square root, the conjugate multiplication method would be a natural choice, except that that method is used for fraction functions. Well, just put
over the number 1 and, voilà, you’ve got a fraction:
Now do the conjugate multiplication.

Multiply the numerator and denominator by the conjugate of
and simplify.

Now substitution works.
which confirms the calculator answer.