Limits

[car0le_la]

Lim (b-x) is
x→b √(x)-√(b)

a)-2√(b)
b)-√(b)
c)√(b)
d)2√(b)


My work:
Lim (b-x) * √(x)+√(b)
x→b √(x)-√(b) √(x)+√(b)
Lim (b-x) (√(x)+√(b))
x→b x+b

But I get lost along this path and don't know how to continue to attain one of the answers.
The answer is supposed to be A but I don't know how to get to it.

 

[Deleted member 4993]

Lim (b-x) is
x→b √(x)-√(b)

a)-2√(b)
b)-√(b)
c)√(b)
d)2√(b)


My work:
Lim (b-x) * √(x)+√(b)
x→b √(x)-√(b) √(x)+√(b)
Lim (b-x) (√(x)+√(b))
x→b x+b

But I get lost along this path and don't know how to continue to attain one of the answers.
The answer is supposed to be A but I don't know how to get to it.

\(\displaystyle \lim_{x \to b} \frac{b - x}{\sqrt{(x)} - \sqrt{(b)}}\)

=\(\displaystyle \lim_{x \to b} \frac{(\sqrt{b})^2 - (\sqrt{x})^2}{\sqrt{(x)} - \sqrt{(b)}}\)

Now in the numerator - you have "difference of square" quantities. Factorize that using:

\(\displaystyle a^2 - b^2 = (a-b)*(a+b)\)

 

[pka]

Note that \(\dfrac{b-x}{x-b}=-1\)

 

[lev888]

(√(x)-√(b)) (√(x)+√(b)) = ?

 

[HallsofIvy]

Lim (b-x) is
x→b √(x)-√(b)

a)-2√(b)
b)-√(b)
c)√(b)
d)2√(b)


My work:
Lim (b-x) * √(x)+√(b)
x→b √(x)-√(b) √(x)+√(b)
Lim (b-x) (√(x)+√(b))
x→b x+b

First, your denominator is wrong.
$(\sqrt{x}-\sqrt{b})(\sqrt{x}+\sqrt{b})$ is x- b, not x+ b.
And $\frac{(b- x)(\sqrt{x}+ \sqrt{b})}{x- b}= \frac{-(x- b)(\sqrt{x}+\sqrt{b})}{x- b}= -(\sqrt{x}+ \sqrt{b}$.
And the limit of that, as x goes to b, is just $\sqrt{b}+ \sqrt{b}= 2\sqrt{b}$.

 

[car0le_la]

Lim (b-x)
x→b √(x)-√(b)
Lim (√(b))^2-(√(x))^2
x→b √(x)-√(b)
Lim (√(b)-√(x)) (√(b)+√(x))
x→b √(x)-√(b)
Lim (-1) (-√(b)+√(x)) (√(b)+√(x))
x→b √(x)-√(b)

From this step we can cross out the bottom and top

Lim (-1) (√(b)+√(x))
x→b
And then proceed to plug in
=(-1) (√(b)+√(b))
=(-1) (2√(b))
=(-2√(b))

Thanks for the help!!