limit when x —> 0 of [sqrt(x+9) – 3] / [sqrt(x+16) – 4]

And what properties should I use. Please show!

April 18, 2016
# How to solve this limit problem?

limit when x —> 0 of [sqrt(x+9) – 3] / [sqrt(x+16) – 4]

And what properties should I use. Please show!

the limit is 4/3 or 1.33333

This goes to 0/0 so it’s a candidate for l’Hopital’s Rule, which says to take the derivative of the numerator / derivative of the denominator.

So the limit is the same as the limit of

.5(x+9)^-.5 / .5(x+16)^-.5

.5’s cancel, and flip it upside down to get

?(x+16) / ?(x+9) which is just 4/3.

Here’s are a few terms from Excel, starting from 1 and then dividing by -2 so that we look at limits from both sides:

11.318198574

-0.51.341735476

0.251.329353094

-0.1251.335376999

0.06251.332325085

-0.031251.333840828

0.0156251.333080432

-0.00781251.333459995

0.003906251.333270055

-0.0019531251.333364986

0.0009765631.333317511

-0.0004882811.333341246

0.0002441411.333329377

-0.000122071.333335311

6.10352E-051.333332344

-3.05176E-051.333333828

1.52588E-051.333333086

h(x) = [?(x + 9) – 3] / [?(x + 16) – 4]

Do you know the l’Hôpital’s rule

Lim f(x) / g(x) = Lim f'(x) / g'(x)

x ? a?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ?? ?? ?? ????x ? a

????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

f(x) = ?(x + 9) – 3

f(x) = [(x + 9)^(1/2)] – 3

f'(x) = (1/2) * (1) * (x + 9)^[(1/2) – 1] – 0

f'(x) = (1/2) * (x + 9)^(- 1/2)

f'(x) = (1/2) / (x + 9)^(1/2)

f'(x) = (1/2) / ?(x + 9)

g(x) = [?(x + 16) – 4

g(x) = [(x + 16)^(1/2)] – 4

g'(x) = (1/2) * (1) * (x + 16)^[(1/2) – 1] – 0

g'(x) = (1/2) * (x + 16)^(- 1/2)

g'(x) = (1/2) / (x + 16)^(1/2)

g'(x) = (1/2) / ?(x + 16)

f'(x) / g'(x) = [(1/2) / ?(x + 9)] / [(1/2) / ?(x + 16)]

f'(x) / g'(x) = [1/?(x + 9)] / [1/?(x + 16)]

f'(x) / g'(x) = [?(x + 16)] / [?(x + 9)]

f'(x) / g'(x) = ?[(x + 16)/(x + 9)]

Lim [?(x + 9) – 3] / [?(x + 16) – 4] = Lim ?[(x + 16)/(x + 9)] = ?(16/9) = 4/3

x ? 0??? ?? ?? ?? ???? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ???? ?? ?? ?? ?? ?????? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ?? ???? ?? ?? ?? ?? ????x ? – 4