How to Solve Limits at Infinity by Using Algebra

Yes, you can solve a limit at infinity using a calculator, but all things being equal, it’s better to solve the problem algebraically, because then you have a mathematically airtight answer. For example, with the problem,

image0.png

the calculator answer of 0.5 is very convincing, but it’s not mathematically rigorous, so if you stop there, the math police may get you.

  1. Try substitution — always a good idea.

    No good. You get ∞ – ∞, which tells you nothing. On to plan B.

    image1.png
  2. Multiply the numerator and denominator by the conjugate of

    image2.png

    and simplify.

    image3.png
  3. Now substitution does work.

    image4.png
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