Find the following limit where f(x) = 4x^2 − 3x.

Question

Find the following limit where f(x) = 4x^2 − 3x.

limit-->(f(x+ delta x)-f(x))/(delta x)
delta x

**delta x means the triangle symbol with x next to it***

Answer 1

what is delta x approching? if it is approching 0 this is the definition of the derivative. so it will be 8x-3. I got this by using the rule that the derivative of ax^n=nax^n-1. but they probboly want you to do it without taking the derivative.so lets start by writing it out.

y=delta*x you don't have to do this I just don't want to keep writing delta*x

(4(x+y)^2-3(x+y)-(4x^2-3x))/(y)

next we want to rewrite this so we can take the limit.

sqrare the x+y
distribute the 4
distribute the 3
get rid of some terms that cancal out and you should get 8*x+4*y-3.

now all we have to do is take the limmit as y approches 0.

i get 8x+4*0-3=8*x-3.

this verifys that the derivative of 4*x^2-3x=8*x-3

Answer 2

To find the limit of the expression (f(x+Δx) - f(x))/(Δx) as Δx approaches 0, we need to substitute the value of f(x) = 4x^2 - 3x into the expression and simplify it.

Let's start by finding f(x+Δx):

f(x+Δx) = 4(x+Δx)^2 - 3(x+Δx)

Expanding the square and simplifying:

f(x+Δx) = 4(x^2 + 2xΔx + (Δx)^2) - 3x - 3Δx

Next, substitute f(x) = 4x^2 - 3x and f(x+Δx) = 4(x^2 + 2xΔx + (Δx)^2) - 3x - 3Δx into the expression:

(f(x+Δx) - f(x))/(Δx) = [(4(x^2 + 2xΔx + (Δx)^2) - 3x - 3Δx) - (4x^2 - 3x)]/(Δx)

Simplifying further:

(f(x+Δx) - f(x))/(Δx) = [4x^2 + 8xΔx + 4(Δx)^2 - 3x - 3Δx - 4x^2 + 3x]/(Δx)

Combining like terms:

(f(x+Δx) - f(x))/(Δx) = (8xΔx + 4(Δx)^2 - 3Δx)/(Δx)

Now, factoring out Δx from the numerator:

(f(x+Δx) - f(x))/(Δx) = Δx(8x + 4Δx - 3)/(Δx)

Notice that we can cancel out Δx from the numerator and denominator:

(f(x+Δx) - f(x))/(Δx) = 8x + 4Δx - 3

Finally, as Δx approaches 0, the term 4Δx becomes negligible:

lim(Δx→0) (f(x+Δx) - f(x))/(Δx) = lim(Δx→0) (8x + 4Δx - 3)

Evaluating this limit, we find that the term 4Δx goes to 0 as Δx approaches 0, and we are left with:

lim(Δx→0) (f(x+Δx) - f(x))/(Δx) = 8x - 3

Therefore, the limit of the expression (f(x+Δx) - f(x))/(Δx) as Δx approaches 0 is 8x - 3.

Find the following limit where f(x) = 4x^2 − 3x.

what is delta x approching? if it is approching 0 this is the definition of the derivative. so it will be 8x-3. I got this by using the rule that the derivative of ax^n=n*a*x^n-1. but they probboly want you to do it without taking the derivative.so lets start by writing it out.

To find the limit of the expression (f(x+Δx) - f(x))/(Δx) as Δx approaches 0, we need to substitute the value of f(x) = 4x^2 - 3x into the expression and simplify it.

what is delta x approching? if it is approching 0 this is the definition of the derivative. so it will be 8x-3. I got this by using the rule that the derivative of ax^n=nax^n-1. but they probboly want you to do it without taking the derivative.so lets start by writing it out.