# How do you find the first two nonzero terms in Maclaurin's Formula and use it to approximate #f(1/3)# given #f(x)=int_0^x sin(t^2) dt# ?

Note: mathematical vigor will show that in some cases we can do this, and in this specific case the method is valid. However, this assumes that we can actually replace the integrand with an infinite power series and the integral remain valid.

Term by term, we now integrate to obtain:

Thus, the initial pair of non-zero terms is:

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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