lim(sqrt(x^2 +1)-sqrt(2))/(x-1) x approaches 1

Arectemieryf0 2022-07-27 Answered
lim ( x 2 + 1 2 ) / ( x 1 )
x approaches 1
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Answers (1)

jbacapzh
Answered 2022-07-28 Author has 18 answers
lim x 1 x 2 + 1 2 x 1 =
Evaluating gives 0/0, indicating a common factor of x-1. So we need to manipulate this a little to exposethis, so we can cancel it and then try to evaluate the limit. Multiplying the top and bottom by the conjugate
lim x 1 x 2 + 1 2 x 1 = lim x 0 x 2 + 1 2 x 1 x 2 + 1 + 2 x 2 + 2 + 2
Simplifying
lim x 1 x 2 + 1 2 x 1 = lim x 0 x 2 + 1 2 ( x 1 ) [ x 2 + 1 + 2 ]
lim x 1 x 2 + 1 2 x 1 = lim x 0 x 2 1 ( x 1 ) [ x 2 + 1 + 2 ]
lim x 1 x 2 + 1 2 x 1 = lim x 0 ( x 1 ) ( x + 1 ) ( x 1 ) [ x 2 + 1 + 2 ]
And there it is, canceling
lim x 1 x 2 + 1 2 x 1 = lim x 0 x + 1 x 2 + 1 + 2
Evaluating
lim x 1 x 2 + 1 2 x 1 = 1 + 1 1 + 1 + 2
lim x 1 x 2 + 1 2 x 1 = 2 2 + 2
lim x 1 x 2 + 1 2 x 1 = 2 2 + 2
lim x 1 x 2 + 1 2 x 1 = 1 2
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