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Evaluate the limit lim x→[infinity]√x²-5/√5x²+1.?
A) 1
B) √5
C) -√5
D) 0

1 Answer

7 votes

The limit as x approaches infinity of the expression √x²-5/√5x²+1 simplifies to √5 after dividing the numerator and denominator by the highest power of x present.

To evaluate the limit lim x→∞ √x²-5/√5x²+1, we divide the numerator and denominator by the highest degree of x in the denominator.

So the expression becomes lim x→∞ (√x²/x)√(1-5/x²)/(√5x²/x)√(1+1/5x²). Since √x²/x simplifies to 1 for positive x and 1-5/x² and 1+1/5x² approach 1 as x approaches infinity, the expression simplifies to lim x→∞ √1/√5, which equals 1/√5.
Consequently, answer is √5.

The square root simplification shows that dividing by the highest power of x helps in finding limits for functions involving infinity, similar to asymptotes described in calculus.

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