Find the following limit.limx→9 of x−9/sq root of (x-3)-sq root of 6
To find the limit of this expression as x approaches 9, we will start by simplifying the expression
To find the limit of this expression as x approaches 9, we will start by simplifying the expression.
We have:
lim(x → 9) [(x – 9) / (√(x – 3) – √6)]
To simplify the expression, we will multiply both the numerator and denominator by the conjugate of the denominator (√(x – 3) + √6).
lim(x → 9) [(x – 9) * (√(x – 3) + √6)] / [ (√(x – 3) – √6) * (√(x – 3) + √6) ]
Using the difference of squares rule, we have:
lim(x → 9) [(x – 9) * (√(x – 3) + √6)] / [(x – 3) – 6]
Simplifying further:
lim(x → 9) [√(x – 3) + √6)]
Now we can substitute x = 9 into the expression:
[√(9 – 3) + √6]
[√6 + √6]
2√6
Therefore, the limit of the given expression as x approaches 9 is 2√6.
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