How to Solve Limits at Infinity Using Algebra - dummies

How to Solve Limits at Infinity Using Algebra

By Mark Ryan

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

image0.png

  1. Try substitution — always a good idea.

    No good. You get

    image1.png

    which tells you nothing. On to plan B.

    Remember that

    image2.png

    does not equal zero. Substitution does not work for this problem. If you plug

    image3.png

    into x, you get

    image4.png

    which does not necessarily equal zero

    image5.png

    As a result,

    image6.png

    tells you nothing about the answer to a limit problem.

    Because

    image7.png

    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

    image8.png

    over the number 1 and, voilà, you’ve got a fraction:

    image9.png

    Now do the conjugate multiplication.

  2. Multiply the numerator and denominator by the conjugate of

    image10.png

    and simplify.

    image11.png

  3. Now substitution works.

    image12.png

    which confirms the calculator answer.