lim_(x rarr+oo)(sqrtxln(x))/(x^(2)+1)=?

Recognize growth rates: First, recognize the growth rates of functions involved. The numerator sqrt(x) * ln(x) grows slower than the denominator x^2. Rewrite expression: Rewrite the expression for clarity: (sqrt(x) * ln(x)) / (x^2 + 1) = (x^(1/2) * ln(x)) / (x^2 + 1). Simplify by division: Simplify the expression by dividing each term in the numerator by x^2: (x^(1/2) / x^2) * ln(x) = (1 / x^(3/2)) * ln(x). Consider limits at infinity: As x approaches infinity, 1/x^(3/2) approaches 0. Consider the behavior of ln(x) as x approaches infinity; ln(x) increases, but at a slower rate compared to any power of x. Multiply the limits: Multiply the two limits: since 1/x^(3/2) approaches 0 and ln(x) approaches infinity, the product (1 / x^(3/2)) * ln(x) approaches 0.

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Calculus

Find derivatives using the chain rule I

Full solution

\lim _{x \rightarrow+\infty} \frac{\sqrt{x} \ln (x)}{x^{2}+1}=?

Recognize growth rates: First, recognize the growth rates of functions involved. The numerator $\sqrt{x} \cdot \ln(x)$ grows slower than the denominator $x^2$ .
Rewrite expression: Rewrite the expression for clarity: $(\sqrt{x} * \ln(x)) / (x^2 + 1) = (x^{1/2} * \ln(x)) / (x^2 + 1)$ .
Simplify by division: Simplify the expression by dividing each term in the numerator by $x^2$ : $(x^{1/2} / x^2) \cdot \ln(x) = (1 / x^{3/2}) \cdot \ln(x)$ .
Consider limits at infinity: As $x$ approaches infinity, $\frac{1}{x^{3/2}}$ approaches $0$ . Consider the behavior of $\ln(x)$ as $x$ approaches infinity; $\ln(x)$ increases, but at a slower rate compared to any power of $x$ .
Multiply the limits: Multiply the two limits: since $\frac{1}{x^{3/2}}$ approaches $0$ and $\ln(x)$ approaches infinity, the product $\left(\frac{1}{x^{3/2}}\right) * \ln(x)$ approaches $0$ .