Question

f(x)=(6x^2)-2x

Find the limit. Deltax->0

Answer 1

\[f \prime (x)=\lim_{ \Delta x \rightarrow 0} f(x+\Delta x)-f(x)/\Delta x=\lim_{\Delta x \rightarrow 0} 6(x+\Delta x)^2-2(x+\Delta x)-(6x^2-2x)/\Delta x\]

Answer 2

Nice just give them the answer...

Answer 3

I've gotten that far but I don't understand how to factor it.

Answer 4

I understand you put it all over deltax but yeah

Answer 5

like that entire equation divided by deltax

Answer 6

@AAbomosalam1998 try to get used to using the fraction command: \frac{}{}
it will make your equations look a lot nicer :)

Answer 7

Yes, that's right, after expanding the brackets and combining like terms, factor out delta x and then take the limit as delta x approaches zero. @zepdrix okay, I am sorry I am not professional with typing out the commands in latex, I will use that next time.

Answer 8

I don't know how to factor a problem with deltax in it, like squaring it, are there rules to it?

Answer 9

6(x+Δx)^2=6(x+Δx)(x+Δx) Multiply each term in the first bracket with each term in the second bracket, add like terms then multiply each term by 6. Do the same to -2(x+Δx).
6x^2+12xΔ x-6Δx^2-2x-2Δx-6x^2+2x=12xΔx-6Δx^2-2Δx . Factor out Δx =12x-6Δ x-2. Take the limit as Δx-->0

Answer 10

\[6x^2+12x \Delta x-6 \Delta x^2-2x-2\Delta x-6x^2+2x=12x \Delta x -6 \Delta x^2-2\Delta x\]
\[ \frac{12x \Delta x-6 \Delta x ^2-2\Delta x}{\Delta x}=\frac{\Delta x(12x-6 \Delta x-2)}{\Delta x}=12x-6\Delta x-2\]

Answer 11

so the limit if x approaches 0 is -2?

Answer 12

if the question is what is the limit as x ->0 of the expression
6x^2 -2x

then the limit will be the sum of the limits of each term
lim (6x^2) + lim(-2x)

the limit of a product is the limit of the products
so
lim(6x^2) = lim(6) * lim(x^2)
the limit of a constant is the constant:
lim(6)*lim(x^2) = 6 * lim(x^2)
any idea what
\[ \lim_{x\rightarrow 0} x^2 \]
?
it is 0

ditto for
lim(-2x) = -2*lim(x)

Answer 13

First of all, we need clarification regarding what you're trying to do here.

f(x)=(6x^2)-2x

Find the limit. Deltax->0 This is vague, because there's no "delta x" in the given f(x).

Are you, by any chance, being asked to find the derivative of f(x)-(6x^2)-2x using the definition of the derivative?

Answer 14

Yes, i am.