Question

Find the limit. Please show all workings.

Answer 1

2x-2

Explanation:

let d = delta x

lim as d goes to 0 ( ( x+d)^2 - 2(x+d) +1 - ( x^2 -2x+1))/d

Expand the term in side the parentheses

( ( x+d)^2 - 2(x+d) +1 - ( x^2 -2x+1))

x^2 +2dx +d^2 -2x-2d+1 - x^2 +2x -1

Combine like terms

2dx +d^2 -2d

Replace

lim as d goes to 0 ( 2dx +d^2 -2d)/d

Factor out a d

lim as d goes to 0 d( 2x +d -2)/d

Cancel the d in the numerator and denominator

lim as d goes to 0 ( 2x +d -2)

Take the limit of each term as d goes to zero

2x +0-2

2x-2

Answer 2

`\displaystyle \lim_(\Delta x \to 0) ((x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1))/(\Delta x)=2x-2`

Explanation:

We want to find the limit:

`\displaystyle \lim_(\Delta x \to 0) ((x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1))/(\Delta x)`

We can expand the numerator:

`=\displaystyle \lim_(\Delta x \to 0) ((x^2+2x\Delta x+\Delta x^2)+(-2x-2\Delta x)+1+(-x^2+2x-1))/(\Delta x)`

Simplify. Combine like terms:

`\displaystyle \lim_(\Delta x\to 0) ((x^2-x^2)+(-2x+2x)+(1-1)+(2x\Delta x+\Delta x^2-2\Delta x))/(y)`

The first three terms will cancel:

`\displaystyle \lim_(\Delta x\to 0) (2x\Delta x+\Delta x^2-2\Delta x)/(\Delta x)`

Factor:

`\displaystyle \lim_(\Delta x \to 0) (\Delta x(2x+\Delta x-2))/(\Delta x)`

Cancel:

`\displaystyle \lim_(\Delta x\to 0)2x+\Delta x-2`

Now, we can use direct substitution:

`\displaystyle \begin{aligned} &\Rightarrow 2x+(0)-2\\ &=2x-2\end{aligned}`

Therefore:

`\displaystyle \lim_(\Delta x \to 0) ((x+\Delta x)^2-2(x+\Delta x)+1-(x^2-2x+1))/(\Delta x)=2x-2`