Find the following limit where f(x) = 4x2 − 3x.
limit
delta x --> f(x + delta x)- f(x)/delta x
can someone please help me
I assume you mean the limit as delta x -> 0.
I'll use delta x = h for simplicity
(f(x + h) - f(x))/h
= (4(x+h)^2 - 3(x+h) - (4x^2 - 3x))/h
= (4x^2 + 8xh + 4h^2 - 3x - 3h - 4x^2 + 3x)/h
= (8xh + 4h^2 - 3h)/h
= 8x - 3 + 4h
Limit as h->0 = 8x-3
Looks like the derivative, eh?
To find the limit of the given expression, we can start by finding the difference quotient, which represents the derivative of the function f(x):
delta x --> [f(x + delta x) - f(x)] / delta x
Here, f(x) = 4x^2 - 3x. We need to substitute this expression into the difference quotient:
delta x --> [(4(x + delta x)^2 - 3(x + delta x)) - (4x^2 - 3x)] / delta x
Next, we can expand and simplify the expression:
delta x --> [(4(x^2 + 2x(delta x) + (delta x)^2) - 3x - 3(delta x)) - (4x^2 - 3x)] / delta x
Simplifying further:
delta x --> [(4x^2 + 8x(delta x) + 4(delta x)^2 - 3x - 3(delta x)) - 4x^2 + 3x] / delta x
Grouping like terms:
delta x --> [8x(delta x) + 4(delta x)^2 - 3(delta x)] / delta x
Now, we can cancel out the terms where delta x is in the denominator and simplify the expression:
delta x --> 8x + 4(delta x) - 3
As delta x approaches zero, the remaining expression becomes:
lim (delta x --> 0) (8x + 4(delta x) - 3)
Now we can substitute delta x = 0 into the expression:
8x + 4(0) - 3
Simplifying further:
8x - 3
Therefore, the limit of the given expression is 8x - 3.