What is the limit as x approaches infinity of (sqrt(x^2+3))/(sqrt(3x^2+1)) ?

The limit as x approaches infinity of (sqrt(x^2+3))/(sqrt(3x^2+1)) is sqrt(1/3)

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limit as x approaches infinity of (sqrt(x^2+3))/(sqrt(3x^2+1))

x \to \infty lim (\frac{x ^{2} + 3}{3 x ^{2} + 1})

\frac{1}{3}

Decimal

0.57735 \dots

Solution steps

x \to \infty lim (\frac{x ^{2} + 3}{3 x ^{2} + 1})

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= x \to \infty lim (\frac{x ^{2} + 3}{3 x ^{2} + 1})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

With the exception of indeterminate form

= x \to \infty lim (\frac{x ^{2} + 3}{3 x ^{2} + 1})

Divide by highest denominator power:

\frac{1 + \frac{3}{x ^{2}}}{3 + \frac{1}{x ^{2}}}

= x \to \infty lim (\frac{1 + \frac{3}{x ^{2}}}{3 + \frac{1}{x ^{2}}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( 1 + \frac{3}{x ^{2}} )}{lim _{x \to \infty} ( 3 + \frac{1}{x ^{2}} )}

x \to \infty lim (1 + \frac{3}{x ^{2}}) = 1

x \to \infty lim (3 + \frac{1}{x ^{2}}) = 3

= \frac{1}{3}

\int \frac{2 x ^{3} + x ^{2} - 5}{x - 1} d x

(x^{2} + 1) y^{^{'}} + 5 x (y - 1) = 0, y (0) = 9

n = 0 \sum \infty q^{n}

\int \frac{1 + 8 v}{- 16 v ^{2}} d v

x \frac{d y}{d x} = x^{2} + 3 y

What is the limit as x approaches infinity of (sqrt(x^2+3))/(sqrt(3x^2+1)) ?
The limit as x approaches infinity of (sqrt(x^2+3))/(sqrt(3x^2+1)) is sqrt(1/3)