What is the limit as x approaches-infinity of (sqrt(4x^2+1))/(3x-1) ?

The limit as x approaches-infinity of (sqrt(4x^2+1))/(3x-1) is -2/3

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limit as x approaches-infinity of (sqrt(4x^2+1))/(3x-1)

x \to - \infty lim (\frac{4 x ^{2} + 1}{3 x - 1})

- \frac{2}{3}

Decimal Notation

- 0.66666 \dots

Solution steps

x \to - \infty lim (\frac{4 x ^{2} + 1}{3 x - 1})

Divide by highest denominator power:

\frac{\frac{4 x ^{2} + 1}{x}}{3 - \frac{1}{x}}

= x \to - \infty lim (\frac{\frac{4 x ^{2} + 1}{x}}{3 - \frac{1}{x}})

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} ( \frac{4 x ^{2} + 1}{x} )}{lim _{x \to - \infty} ( 3 - \frac{1}{x} )}

x \to - \infty lim (\frac{4 x ^{2} + 1}{x}) = - 2

x \to - \infty lim (3 - \frac{1}{x}) = 3

= \frac{- 2}{3}

Simplify

= - \frac{2}{3}

x \to 2 - lim (e^{\frac{1}{( x - 2 )}})

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

x \to 3 + lim (\frac{1 - x}{x - 3})

x \to 1 - lim (x^{\frac{1}{( 1 - x )}})

x \to \infty lim (1 - x^{3})

What is the limit as x approaches-infinity of (sqrt(4x^2+1))/(3x-1) ?
The limit as x approaches-infinity of (sqrt(4x^2+1))/(3x-1) is -2/3