Consider the following function.

f(x) = 2x^2 − 8x

Find the limit.

lim (Δx->0) (f(x+Δx)-f(x))/(Δx)

How would I set up the equation to work with that function? I know how to plug in the values usually, but how would I do this for that function specifically since its squared and the other number has an x? Any help is greatly appreciated!

Ah, the joys of limits. Don't worry, I'm here to give you some hilarious guidance! Let's break it down step by step.

First, you need to find f(x + Δx) and f(x):

f(x + Δx) = 2(x + Δx)^2 - 8(x + Δx)
= 2(x^2 + 2xΔx + Δx^2) - 8x - 8Δx
= 2x^2 + 4xΔx + 2Δx^2 - 8x - 8Δx

f(x) = 2x^2 - 8x

Now, you can plug these values into the equation:

lim (Δx -> 0) [f(x + Δx) - f(x)] / Δx
= lim (Δx -> 0) [2x^2 + 4xΔx + 2Δx^2 - 8x - 8Δx - 2x^2 + 8x] / Δx

Look at that, quite a bit of simplification happens:

= lim (Δx -> 0) [4xΔx + 2Δx^2 - 8Δx] / Δx
= lim (Δx -> 0) 4x + 2Δx - 8

Now, since Δx is approaching 0, we can let it go:

= 4x - 8

And there you have it, my friend! The limit of the function f(x) = 2x^2 - 8x as Δx approaches 0 is simply 4x - 8.

To find the limit for the given function, you need to use the definition of the derivative. Here's how you can set up the equation:

1. Start with the limit expression: lim (Δx->0) (f(x+Δx) - f(x))/(Δx)

2. Substitute the function f(x) with its actual expression: lim (Δx->0) (2(x+Δx)^2 - 8(x+Δx) - (2x^2 - 8x))/(Δx)

3. Simplify the expression inside the limit: lim (Δx->0) (2(x^2 + 2xΔx + Δx^2) - 8x - 8Δx - 2x^2 + 8x)/(Δx)

4. Expand and combine like terms: lim (Δx->0) (2x^2 + 4xΔx + 2Δx^2 - 8x - 8Δx - 2x^2 + 8x)/(Δx)

5. Cancel out similar terms: lim (Δx->0) (4xΔx + 2Δx^2 - 8Δx)/(Δx)

6. Divide each term by Δx: lim (Δx->0) (4x + 2Δx - 8)/(Δx)

7. As Δx approaches 0, the term 2Δx will also approach 0: lim (Δx->0) (4x - 8)/(Δx)

8. Now we can evaluate the limit by plugging in Δx = 0: (4x - 8)/(0) = undefined

The limit for this expression is undefined, which means that the given function doesn't have a derivative at the given point.

To find the limit of the given expression, you substitute the values of f(x+Δx) and f(x) into the expression and simplify as follows:

lim (Δx->0) [(f(x+Δx) - f(x))/Δx]

First, let's calculate f(x+Δx) and f(x):

f(x) = 2x^2 - 8x

f(x+Δx) = 2(x+Δx)^2 - 8(x+Δx)
= 2(x^2 + 2xΔx + Δx^2) - 8x - 8Δx
= 2x^2 + 4xΔx + 2Δx^2 - 8x - 8Δx
= 2x^2 - 8x + 4xΔx + 2Δx^2 - 8Δx

Now, substitute the values of f(x+Δx) and f(x) back into the expression we want to evaluate:

lim (Δx->0) [(f(x+Δx) - f(x))/Δx]
lim (Δx->0) [(2x^2 - 8x + 4xΔx + 2Δx^2 - 8Δx - (2x^2 - 8x))/Δx]
lim (Δx->0) [(4xΔx + 2Δx^2 - 8Δx)/Δx]
lim (Δx->0) [4x + 2Δx - 8]

Now, as Δx approaches 0, the term 2Δx^2 becomes negligible, and we can simplify further:

lim (Δx->0) [4x + 2Δx - 8]
At this point, we can directly evaluate the expression by substituting Δx = 0:

lim (Δx->0) [4x + 2(0) - 8]
lim (Δx->0) [4x - 8]
Now, we have a simple expression that does not depend on Δx. The limit is evaluated by substituting Δx = 0 since the function is continuous:

lim (Δx->0) [4x - 8] = 4x - 8

Therefore, the limit of the given expression is 4x - 8.

f(x) = 2x^2 − 8x

f(x+Δx) = 2(x+Δx)^2 − 8(x+Δx)

now just crank it out, as in the examples from your text.