? asked in 科學及數學數學 · 1 decade ago

limit

can anybody explain why lim(x->0) sin(x)/x = 1 to me please?

and also, the explanation for lim(x->infinity) sin2x/x= 0.

i want the proofs, explanations, and all procedures for calculating these.

in addition, how to use definition of limit to prove the limt to be exist or not exist? give me some examples in details.

1 Answer

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  • 1 decade ago
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    First question:

    With reference to the diagram below:

    圖片參考:http://i117.photobucket.com/albums/o61/billy_hywun...

    Second question: Again, it can be proved by the squeezing principle:

    圖片參考:http://i117.photobucket.com/albums/o61/billy_hywun...

    Third question: Existence of limit can be found by verifying the existence and consistency of left-hand and right-hand limits, for example:

    圖片參考:http://i117.photobucket.com/albums/o61/billy_hywun...

    So from the result, the limit of x - [x] does not exist for x tends to 2 since the left-hand and right hand limits are not equal.

    In fact for x - [x], its limit does not exist for all x tends to any integer (either positive or negative).

    Therefore, the limit of a function can be said to be not existing if:

    Either left-hand or right-hand or both limits do not exist; OR

    Both limits exist but they are NOT equal to each other

    Source(s): My Maths knowledge
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