# How to expand cos(x+h) in powers of x and h?

For functions with two variables, the Taylor expansion's general expression is

so

but

so

Knowing that from is another way to accomplish that.

Managing multivariate series requires a great deal of work due to the necessary notation.

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To expand cos(x+h) in powers of x and h, we can use the Taylor series expansion. The Taylor series expansion for cos(x) is:

cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...

To expand cos(x+h), we substitute (x+h) in place of x in the above series:

cos(x+h) = 1 - ((x+h)^2)/2! + ((x+h)^4)/4! - ((x+h)^6)/6! + ...

Expanding each term using the binomial theorem, we simplify the expression:

cos(x+h) = 1 - (x^2 + 2xh + h^2)/2! + (x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4)/4! - ...

Simplifying further, we collect like terms and arrange them in powers of x and h:

cos(x+h) = 1 - (x^2)/2! + (x^4)/4! - ... + (-1)^n * (x^(2n))/((2n)!) + ... - (xh)/2! + (4x^3h)/4! - ... + (-1)^n * ((2n-1)x^(2n-1)h)/((2n)!) + ... + (h^2)/2! - (6x^2h^2)/4! + ... + (-1)^n * ((2n-2)x^(2n-2)h^2)/((2n)!) + ... - (x^3h)/2! + (4xh^3)/4! - ... + (-1)^n * ((2n-1)x^(2n-1)h^3)/((2n)!) + ... + (h^4)/2! - (6x^2h^4)/4! + ... + (-1)^n * ((2n-2)x^(2n-2)h^4)/((2n)!) + ...

This expansion can be continued indefinitely, including terms involving higher powers of x and h.

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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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