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Useful limit of a function with cosine

Krystian Karczyński

Founder and General Manager of eTrapez.

Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.

He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.


We usually treat the limits of a sine function as: (you can find the derivation of this formula here ).

Question: what about cosine of x?

Formula for the limit of a function with cosine

Of course, this does not happen: , because the limit of the function it is not an unmarked symbol at all.

For limits on the cosine of x, the formula is often helpful:

– in many textbooks it is already given “at the beginning”, without any demonstration, and in many textbooks it is given as a limit of a function that needs to be calculated.

Regardless of your case, it is worth knowing how to derive this formula, and it goes like this:

At this point, I use trigonometric one in the numerator:

The limit of the function z runs down to , because:

And the limit of the function z runs down to , because , so we have the result:

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