Question

How do I solve limits with delta x

Answer 1

Hi,

Explanation:

49.

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

50.

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

52.

`\displaystyle \lim_(\Delta x \to 0) ((x+\Delta x)^3-x^3)/(\Delta x)\\\\=\lim_(\Delta x \to 0) (x^3+3x^2*\Delta x +3*x*\Delta^2 x+\Delta x^3-x^3 )/(\Delta x) \\\\=\lim_(\Delta x \to 0) (3x^2*\Delta x +3*x*\Delta^2 x+\Delta x^3 )/(\Delta x) \\\\=3x^2+0+0\\\\=3x^2`