Question

Evaluate the limit

Answer 1

Two options. Either multiply top and bottom by the conjugate of the bottom, or take advantage of difference of squares:
\[a-b = (\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})\]

Answer 2

ok i think id rather take the conjugate

Answer 3

Either works :) The difference of squares option isn't as obvious when you're not used to it with things that aren't actually like \(x^{2}-4\). But it can be done with the anything that is a-b form.

Answer 4

the conjugate is \[1+\sqrt{1}\] right?

Answer 5

Top and bottom by \(1+\sqrt{b}\)

Answer 6

ok once we do that do i multiply them out?

Answer 7

Multiply out the bottom one. Multiplying out the top one makes it hard to see what happens.

Answer 8

ok so how does the bottom look like?

Answer 9

1-b?

Answer 10

then that cancels out with the top right ?

Answer 11

\[\frac{ (1-b)(1+\sqrt{b}) }{ (1-\sqrt{b})(1+\sqrt{b}) } = \frac{ (1-b)(1+\sqrt{b}) }{ 1+\sqrt{b}-\sqrt{b}-b } = \frac{ (1-b)(1+\sqrt{b}) }{ 1-b }\]

Answer 12

Yes, then it would cancel :)

Answer 13

then 1+\[\sqrt{b}\]

Answer 14

would be left over right?

Answer 15

Yep. And then you can plug in your limit.

Answer 16

so the answer would be 2

Answer 17

Yes :3

Answer 18

ah yes thank you

Answer 19

You're welcome :)