Hi! How would you find the limit?
limit of (f(x+Δx)-f(x)) / Δ x with Δ x -> 0
f(x)= 3x-2
the Δx confuses me... How do I solve the problem when it has Δx?
this is the definition of the derivative
f(x+dx) = 3(x+dx)-2 = 3 x + 3 dx -2
f(x) = 3 x - 2
f(x+dx) - f(x) = 3 dx
divide by dx
= 3
By the way since this particular function is linear, constant slope, there was no need for us to take the limit to find the derivative (slope at x is slope at any x)
OHH. THANK YOU SO MUCH!!
You are welcome.
To find the limit of the given expression, we need to substitute the function f(x) with the given expression into the limit expression. Let's plug in the function:
limit as Δx -> 0 of [(f(x+Δx) - f(x)) / Δx]
f(x) = 3x - 2
Substitute the values into the expression:
= limit as Δx -> 0 of [(3(x+Δx) - 2) - (3x - 2)] / Δx
Next, simplify the expression:
= limit as Δx -> 0 of (3x + 3Δx - 2 - 3x + 2) / Δx
= limit as Δx -> 0 of (3Δx) / Δx
Now, we notice that Δx is a common factor in the numerator and denominator, and as Δx approaches 0, the expression simplifies to:
= limit as Δx -> 0 of 3
Therefore, the limit of the expression (f(x+Δx) - f(x)) / Δx as Δx approaches 0 is equal to 3.