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Posted 3 months ago
2. limx9x4+1x23x+5\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{4}+1}}{x^{2}-3 x+5} is
(A) 1
(B) 3
(C) 9
(D) nonexistent
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 2
Simplifying the expression, we get limx9+1x413x+5x2 \lim_{x \to \infty} \frac{\sqrt{9 + \frac{1}{x^4}}}{1 - \frac{3}{x} + \frac{5}{x^2}} ..
step 3
As x x approaches infinity, the terms 1x4 \frac{1}{x^4} , 3x \frac{3}{x} , and 5x2 \frac{5}{x^2} approach zero..
step 4
The limit then simplifies to limx91 \lim_{x \to \infty} \frac{\sqrt{9}}{1} ..
step 5
The final result is 3 3 ..
Key Concept
Limit of a Rational Function with a Radical as x x Approaches Infinity
When finding the limit of a rational function with a radical as x x approaches infinity, divide by the highest power of x x in the denominator and simplify. Terms with x x in the denominator approach zero, and the limit can be evaluated by substituting the remaining terms.

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