Question

Find the limit as deltax approaches 0 of-
root((x-deltax)-4)-root(x-4)/deltax

Answer 1

\[\frac{ \sqrt{(x+Deltax)-4}-\sqrt{x-4} }{ Deltax }\]

Answer 2

For my own convenience, I'm going to write \(h\) instead of \(\Delta x\).
\[\lim_{h\to0}\frac{\sqrt{(x+h)-4}-\sqrt{x-4}}{h}\]
Multiply numerator and denominator by the conjugate of the numerator:
\[\lim_{h\to0}\frac{\sqrt{(x+h)-4}-\sqrt{x-4}}{h}\cdot\frac{\sqrt{(x+h)-4}+\sqrt{x-4}}{\sqrt{(x+h)-4}+\sqrt{x-4}}\]
Since \((a+b)(a-b)=a^2-b^2\), you have
\[\lim_{h\to0}\frac{\left[\sqrt{(x+h)-4}\right]^2-\left[\sqrt{x-4}\right]^2}{h\left(\sqrt{(x+h)-4}+\sqrt{x-4}\right)}\\
\lim_{h\to0}\frac{(x+h)-4-\left(x-4\right)}{h\left(\sqrt{(x+h)-4}+\sqrt{x-4}\right)}\\
\lim_{h\to0}\frac{x+h-4-x+4}{h\left(\sqrt{(x+h)-4}+\sqrt{x-4}\right)}\\
\lim_{h\to0}\frac{h}{h\left(\sqrt{(x+h)-4}+\sqrt{x-4}\right)}\\
\lim_{h\to0}\frac{1}{\sqrt{(x+h)-4}+\sqrt{x-4}}\\
\]

Answer 3

so as h approaches 0 we should just get 1/2root(x-4) right?

Answer 4

Yes, you can verify with the power/chain rules, if you've learned them.

Answer 5

I have and I will! thank you!

Answer 6

You're welcome!

Answer 7

I have one more questions actulaly

Answer 8

I solved it myself but im not sure if it was right

Answer 9

The position of a function is given by s(t) = 4t2 + 2t + 3, where s is measured in feet and t in seconds. Find the velocity of the object at time t = 2 seconds. Include units in your answer.

Answer 10

I simply found the first derivative which gave me 8t+2 then plugged in 2 for t. I that right?

Answer 11

Yes, you only have to find \(s'(2)\).

Answer 12

ok thanks! I thought it was a bit easy so I got worried for a sec