Find limit X approaches 1 for
((5-X)^.5 -2)/((2-X)^.5 -1)
e-mail address: mark.hultgren
Thank you.
Substitute x = 1 - t and expand the squareroots in series using the formula:
sqrt[1 + y] = 1 + y/2 + O(y^2)
You should then find that the limit is 1/2
or
Multiply by ((5-X)^.5 + 2)/((5-X)^.5 + 2)*((2-X)^.5 + 1)/((2-X)^.5 + 1)
which reduces your question to
Limit ((2-X)^.5 + 1)/((5-X)^.5 + 2) as x-->1
= 2/4
= 1/2
BTW, it is strongly suggested that you do not put your email or personal information in these postings
To find the limit as X approaches 1 for the given expression, we can use algebraic manipulation and apply the concept of limits.
Let's start by simplifying the expression:
((5 - X)^0.5 - 2)/((2 - X)^0.5 - 1)
To eliminate the square roots in the denominator, we can multiply both the numerator and the denominator by the conjugate of the denominator: ((2 - X)^0.5 + 1):
[((5 - X)^0.5 - 2)/((2 - X)^0.5 - 1)] * [((2 - X)^0.5 + 1)/((2 - X)^0.5 + 1)]
By multiplying the conjugate, we are using the difference of squares formula, which allows us to eliminate the square root in the denominator.
Now, we have:
[(5 - X)^0.5 - 2] * [(2 - X)^0.5 + 1]/[(2 - X) - 1]
Simplifying further:
[(5 - X)^0.5 - 2] * [(2 - X)^0.5 + 1]/(1 - X)
Now, we can rewrite this equation using the difference of squares formula again:
[((5 - X) - 4) * [(2 - X)^0.5 + 1]/(1 - X)
Simplifying the numerator:
(1 - X) * [(2 - X)^0.5 + 1]/(1 - X)
The (1 - X) terms cancel out:
[(2 - X)^0.5 + 1]
Now, we can substitute X = 1 into the expression:
[(2 - 1)^0.5 + 1]
= (1^0.5 + 1)
= 1 + 1
= 2
Hence, the limit as X approaches 1 for the given expression is 2.
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