Question

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Evaluate the limit.
lim ₓ →[infinity]√(x²+3)/√(3 x²+1)

Answer 1

To find the limit of √(x²+3)/√(3x²+1) as x approaches infinity, divide the numerator and the denominator by x, simplify the expression, and take the limit, which results in √3/3.

Step-by-step explanation:

The student is asking to evaluate the limit as x approaches infinity of the expression √(x²+3)/√(3x²+1). To solve this limit, we can divide both the numerator and the denominator by x. This simplifies the expression inside the square roots, highlighting the dominant terms which are x² in the numerator and 3x² in the denominator. As x tends to infinity, the lesser terms (3 and 1) become negligible.

The mathematical process is as follows:

1. Divide both the numerator and the denominator by x.
2. Simplify the square roots.
3. Take the limit as x approaches infinity, which results in the square roots of the leading coefficients 1 and 3 respectively.
4. Realize that the limit tends to √(1/3), which simplifies to 1/√3 or √3/3 after rationalizing the denominator.

Therefore, the limit of the expression as x approaches infinity is √3/3.