6. Laws of Limits

Calculus 1



Weiqi Zhou

Finite Summability

If , ，then

Proof



  • By definition of convergence, for any ，there are ，so that holds for all

  • Thus for we have

Finite Multipliability

If , ，then

Proof



  • There exists ，so that holds for all , since convergence implies boundedness

  • By definition of convergence, there exists ，so that for any we have

  • for large enough

Example

Exercise

  • Suppose that does not exist，check whether following statments are correct, and why

  • If , then still does not exist

  • If does not exist, then neither does

  • If does not exist, then neither does

Solution

  • Correct，if it exists,then would also exist, which is a contradiction

  • Counterexample：, then

  • Counterexample：, then

Nonexamples

Monotonicity

If , ，and , then

Proof

  • Let ，then ，and

  • If ，then there would exist some ，so that holds for all ，which contradicts

  • Therefore ，i.e.,

These properties also carry over to limits of functions

Composition

If , ，then

Exercise

  • If ， is monotonically increasing and unbounded，decide whether following statements are true, and why



  • does not exist

  • does not exist，but exists

Solution

  • Counterexample: ，then

  • Counterexample：: , then

  • Correct, for similar reasons as in the last exercise

NOT Necessarily Commutative

Example：, then

Summary

  • Finite Summability and Finite Multipliability

  • Monotonicity and Composition

  • Non-commutativity