How to Solve Limits at Infinity Using Algebra - dummies

# 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

1. 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,

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.

2. Multiply the numerator and denominator by the conjugate of

and simplify.

3. Now substitution works.