Find the limit as it approaches zero. ( f( x + change in x) - f(x) )/ change in x given f(x)=2x-5.
( change in x = triangle x)
f+dx = 2(x+dx) - 5
= 2 x + 2 dx -5
f(x) = 2 x - 5
((f(x+dx) - f(x)) = 2 dx
divide by dx
= 2
no need to approach 0 with this, the slope of a straight line is constant.
by the way the derivative of 2 x - 5 is 2
To find the limit as it approaches zero of the expression (f(x + Δx) - f(x)) / Δx, where f(x) = 2x - 5, we can substitute the given function and simplify the expression.
Let's start by substituting f(x) into the expression:
[(2(x + Δx) - 5) - (2x - 5)] / Δx
Simplify the expression by distributing the 2:
[(2x + 2Δx - 5) - (2x - 5)] / Δx
Simplify further by combining like terms:
[2x + 2Δx - 5 - 2x + 5] / Δx
Cancel out the 2x terms:
[2Δx] / Δx
The Δx in the numerator and the Δx in the denominator cancel out, leaving us with:
2
Therefore, the limit as it approaches zero of the expression (f(x + Δx) - f(x)) / Δx, given f(x) = 2x - 5, is equal to 2.
To find the limit as it approaches zero of the expression ( f( x + Δx) - f(x) )/ Δx, first substitute the function f(x) = 2x - 5 into the expression:
( f( x + Δx) - f(x) )/ Δx = ( (2(x + Δx) - 5) - (2x - 5) )/ Δx
Simplifying further:
= (2x + 2Δx - 5 - 2x + 5 )/ Δx
= (2Δx)/ Δx
= 2
Therefore, the limit as Δx approaches zero of ( f( x + Δx) - f(x) )/ Δx is equal to 2.