Question

Math

Posted 3 months ago

```
2. $\lim _{x \rightarrow \infty} \frac{\sqrt{9 x^{4}+1}}{x^{2}-3 x+5}$ 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 $\lim_{x \to \infty} \frac{\sqrt{9 + \frac{1}{x^4}}}{1 - \frac{3}{x} + \frac{5}{x^2}}$..

step 3

As $x$ approaches infinity, the terms $\frac{1}{x^4}$, $\frac{3}{x}$, and $\frac{5}{x^2}$ approach zero..

step 4

The limit then simplifies to $\lim_{x \to \infty} \frac{\sqrt{9}}{1}$..

step 5

The final result is $3$..

B

Key Concept

Limit of a Rational Function with a Radical as $x$ Approaches Infinity

Explanation

When finding the limit of a rational function with a radical as $x$ approaches infinity, divide by the highest power of $x$ in the denominator and simplify. Terms with $x$ in the denominator approach zero, and the limit can be evaluated by substituting the remaining terms.

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