Pre calculus, Evaluate the limit

lim[x:-1,(x+1)/(√(x^2+4x+5)-√(x^2+1))]

hey, an additional information-

i havent been taught derivatives yet. We do it by the cancellation-of-the-culprit factor method. so in this case, (x+1)! Please could you do it without using derivatives? thanks so much!

10 points for best answer, I require steps please!

Thankyou so much!

I'd rationalize the denominator

(x + 1) * (sqrt(x^2 + 4x + 5) + sqrt(x^2 + 1)) / (x^2 + 4x + 5 - x^2 - 1) =>
(x + 1) * (sqrt(x^2 + 4x + 5) + sqrt(x^2 + 1)) / (4x + 4) =>
(x + 1) * (sqrt(x^2 + 4x + 5) + sqrt(x^2 + 1)) / (4 * (x + 1)) =>
(sqrt(x^2 + 4x + 5) + sqrt(x^2 + 1)) / 4

x goes to -1

(sqrt((-1)^2 + 4 * (-1) + 5) + sqrt((-1)^2 + 1)) / 4 =>
(sqrt(1 - 4 + 5) + sqrt(1 + 1)) / 4 =>
(sqrt(6 - 4) + sqrt(2)) / 4 =>
(sqrt(2) + sqrt(2)) / 4 =>
2 * sqrt(2) / 4 =>
sqrt(2) / 2

There you go.

hi i am telling you the answer but You have to check once.coz i m also studying.
You can do this by rationlisation.(multiply and devide the denomimator)

(x+1)*[sqrt(x^2 +4x+ 5) +sqrt(x^2+1)] /(x^2+4x+5-x^2 - 1)


(x+1)*[sqrt(x^2+4x+5)+sqrt(x^2+1)]/(4x…


(x+1)*[sqrt(x^2+4x+5)+sqrt(x^2+1)]/[4(… + 1)]



[sqrt(x^2+4x+5)+
sqrt(x^2+1)]/4


then put the limit x:-1 (what is the limit -1 or 1)

if it is 1 then you will get
(sqrt10+sqrt2)/4

if it is -1 you will get
sqrt2/2

lim x—> -1 (x+1)/("" (x^2+4x+5)-√(x^2+1)))
Multiply numerator and denominator with ((x^2+4x+5)+√(x^2+1))
Then the expression becomes
lim x—> -1 ((x+1)("" (x^2+4x+5)+√(x^2+1)))/((x^2+4x+5)-(x^2+1…

which simplifies to
lim x—> -1 ((x+1)("" (x^2+4x+5)+√(x^2+1)))/((4x+4))

or
lim x—> -1 (("" (x^2+4x+5)+√(x^2+1)))/4 = 22/4 = 1/2


I was writing the above in word and cutting and pasteing. the boxes should all read √