I decided to take a look at the Princeton Review book for AB and BC calc and I noticed that the last two problems in the limit review section had multiple variables ie:
lim h -> 0 (sin(x+h)-sin(x))/h

Will these types of problems be on the AB AP Calc test or just BC?

In the AP Calculus Course guide there are sample multiple choice questions. Question one tests on this, except it uses cosine(3/4pi). The exam will try to trick you like this because the limit can't be easily computed, but as long as you know the differentiation rules you'll be fine.

That's not a multivariable limit problem. There is only one variable, x. Do you consider the equation x^2 - x = 0 to have multiple variables? It's the same idea here.

Remember from Algebra I the formula for the average slope of a line ((y2-y1)/(x2-x1))? If y2 = f(x2) and y(1) = f(x1), the average slope of a line is (f(x2) - f(x1))/(x2 - x1).

Let h = x2 - x1 (imagine the numbers after the variables I'm writing are subscripts, write it down on a piece of paper if you have to, it looks really messy on screen I know). h can also be written as delta x (change in x, written as an upper case greek letter delta next to the x), which in this case is the difference between the second and first value of x. If h = x2 -x1, than that means that x2 = x1 + h. So in numerator of the formula for the average slope of the line replace the x2 with x1 + h and in the denominator replace the x2 - x1 with h. This results in (f(x1) + h) - f(x1))/h. If you drop the subscripts it becomes very similar to what you typed in your original post (this is known as the "difference quotient").

h was previously defined as the change in x (aka delta x). If you let the change in x become very small (approaching 0), then you have the limit definition of the derivative, which is what you posted.

## Replies to: Multivariable AB AP Limit Problems?

Remember from Algebra I the formula for the average slope of a line ((y2-y1)/(x2-x1))? If y2 = f(x2) and y(1) = f(x1), the average slope of a line is (f(x2) - f(x1))/(x2 - x1).

Let h = x2 - x1 (imagine the numbers after the variables I'm writing are subscripts, write it down on a piece of paper if you have to, it looks really messy on screen I know). h can also be written as delta x (change in x, written as an upper case greek letter delta next to the x), which in this case is the difference between the second and first value of x. If h = x2 -x1, than that means that x2 = x1 + h. So in numerator of the formula for the average slope of the line replace the x2 with x1 + h and in the denominator replace the x2 - x1 with h. This results in (f(x1) + h) - f(x1))/h. If you drop the subscripts it becomes very similar to what you typed in your original post (this is known as the "difference quotient").

h was previously defined as the change in x (aka delta x). If you let the change in x become very small (approaching 0), then you have the limit definition of the derivative, which is what you posted.