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

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

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

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

\infty

Solution steps

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

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

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

\frac{x ^{3} - 2 x ^{2} + 3}{2 x + 1} = \frac{x ^{3} ( 1 - \frac{2}{x} + \frac{3}{x ^{3}} )}{x ( 2 + \frac{1}{x} )}

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

Cancel the common factor:

x

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

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

With the exception of indeterminate form

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

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

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

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

= 3 \frac{\infty}{2}

Simplify

3 \frac{\infty}{2} : \infty

= \infty

x \to 0 lim (\frac{arcsin ( 5 x )}{sin ( 3 x )})

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

x \to \infty lim (5 x - 2)

h \to 0 lim (\frac{3 h}{h})

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