Final answer:
The limit of the function f(x)=6x+2 as Δx approaches 0 is found to be 6 after simplifying the expression for the difference quotient and observing that the Δx terms cancel out.
Step-by-step explanation:
To find the limit of the function f(x)=6x+2 as Δx approaches 0, we evaluate the expression №(Δx → 0) [(f(x+Δx)-f(x))/Δx]. We substitute the function f(x) into this expression to get:
№(Δx → 0) [(f(x+Δx)-f(x))/Δx] = №(Δx → 0) [(6(x+Δx)+2-(6x+2))/Δx].
Simplifying the numerator gives:
6(x+Δx)+2-(6x+2) = 6Δx.
Dividing both sides by Δx, we get:
№(Δx → 0) [(6Δx)/Δx] = №(Δx → 0) [6].
The Δx terms cancel each other out, so we are left with the constant 6. Therefore, the limit as Δx approaches 0 is simply 6.
Thus, the limit of f(x)=6x+2 as Δx approaches 0 is 6.