# How do you find the Maclaurin series of #f(x)=e^x# ?

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To find the Maclaurin series of ( f(x) = e^x ), you can use the definition of the Maclaurin series, which is given by:

[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n ]

For ( f(x) = e^x ), the derivatives are:

[ f'(x) = e^x ] [ f''(x) = e^x ] [ f'''(x) = e^x ] [ \text{and so on...} ]

Evaluating these derivatives at ( x = 0 ), we find that:

[ f(0) = e^0 = 1 ] [ f'(0) = e^0 = 1 ] [ f''(0) = e^0 = 1 ] [ f'''(0) = e^0 = 1 ] [ \text{and so on...} ]

Thus, the Maclaurin series for ( e^x ) is:

[ e^x = \sum_{n=0}^{\infty} \frac{1}{n!}x^n ]

Or more compactly:

[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} ]

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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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