Question

Consider the following function. f(x) = 1 x 6 find the limit. lim δx→0 f(x δx) − f(x) δx

Answer 1

To find the limit of the function, we will apply the limit definition and simplify the expression. The limit of the given function is 6 / (x(x + δx)^5).

Step-by-step explanation:

To find the limit of the function, we will apply the limit definition. Let's substitute x + δx into the function, subtract f(x), and divide by δx.

limδx→0 [f(x + δx) - f(x)] / δx

Replacing f(x) with the given function 1/x^6, we have:

limδx→0 [1 / (x + δx)^6 - 1 / x^6] / δx

Next, simplify the expression by finding common denominators:

limδx→0 [x^6 - (x + δx)^6] / (x^6(x + δx)^6 δx)

Expanding (x + δx)^6 and canceling out like terms, we get:

limδx→0 [6x5δx + 15x4δx2 + 20x3δx3 + 15x2δx4 + 6xδx5 + δx6] / (x6(x + δx)6δx)

Now we can take the limit as δx approaches 0:

limδx→0 6x5 / (x6(x + δx)5)

Finally, simplify further:

limδx→0 6 / (x(x + δx)5)

This is the limit of the given function. If you have a specific value for x, you can substitute it and evaluate the limit.