Calculus Methods

04 Finding Limits

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        1) Rewrite expression as a single fraction. This step

        is not strictly necessary, but it simplifies a great

        many things.

 

 

        2) Just plug in the limiting value!!

 

        

        3) If you get an answer, you are done. If you get

        zero in only the denominator, your answer is

        "undefined," "no limit," or "unbounded."

        Otherwise, you must have a zero in both the

        numerator and denominator: follow one of the next

        steps:

 

 

        4) If the numerator and denominator are both

        polynomials:

 

                 A) Factor both the numerator and

                denominator.

 

                B) Cancel any terms in common between the

                numerator and denominator.

 

                C) Retake the limit.

 

 

        5) If trigonometric functions are involved, check to

        see if the fraction is of one of the following to forms

        (or can be made to look like one of the following

        two forms): or . Then you must learn the

        following two limits: and .

 

 

        6) For limits where the limiting value goes to

        infinity, please see method 10, Limits at Infinity.

 

        7) For functions of forms different from those

        mentioned in steps 4 and 5, and if you are in BC

        Calculus, see method 28, L'Hopital's Rule.

 

 

        

 

        Example #1: Find the limit: .

 

        1) It's already a single fraction.

 

        2)

 

        3) Since we get an answer, we are finished. The

        answer is -1.

 

 

 

 

        Example #2: Find the limit: .

 

        1) It's already a single fraction.

 

        2)

 

        3) We don't get an answer, so we must move on.

 

        4) The numerator and denominator are not both

        polynomials.

 

        5) The fraction does not immediately look like

        either of the two forms we are supposed to

        remember, so we must make it look like one of

        them.

 

           

 

        We know the answer to this limit problem:

        .

 

 

 

        Example #3: Find the limit: .

 

        1) It's already a single fraction.

 

        2)

 

        3) We don't get an answer, so we must move on.

 

        4) The numerator and denominator are not both

        polynomials.

 

        5) The fraction does not immediately look like

        either of the two forms we are supposed to

        remember, so we must make it look like one of

        them.

 

        First, we multiply the top and bottom by 5 to make

        the argument of the sin function look the same as

        the numerator.

                                   

 

        Next we change the limit from x going to 0 to 5x

        going to 0. We can do that since it's true!

                 

 

            Now we know the limit of the function:

                          

 

 

 

 

        Example #4: Find the limit: .

 

        1)

 

        2)

 

        3) Since we have a zero in the denominator only,

        the answer is "no limit."

 

 

 

        Example #5: Find the limit: .

 

        1) It's already a single fraction.

 

        2)

 

        3) Since we don't get an answer, we move on.

 

        4A) Remembering the factors of , we have

                 

 

        4B)

 

        4C)