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

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

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

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

- 2

Solution steps

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

Expand

\frac{( 2 x - 1 ) ( 3 - x )}{( x - 1 ) ( x + 3 )} : \frac{- 2 x ^{2} + 7 x - 3}{x ^{2} + 2 x - 3}

= x \to \infty lim (\frac{- 2 x ^{2} + 7 x - 3}{x ^{2} + 2 x - 3})

Divide by highest denominator power:

\frac{- 2 + \frac{7}{x} - \frac{3}{x ^{2}}}{1 + \frac{2}{x} - \frac{3}{x ^{2}}}

= x \to \infty lim (\frac{- 2 + \frac{7}{x} - \frac{3}{x ^{2}}}{1 + \frac{2}{x} - \frac{3}{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} ( - 2 + \frac{7}{x} - \frac{3}{x ^{2}} )}{lim _{x \to \infty} ( 1 + \frac{2}{x} - \frac{3}{x ^{2}} )}

x \to \infty lim (- 2 + \frac{7}{x} - \frac{3}{x ^{2}}) = - 2

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

= \frac{- 2}{1}

Simplify

= - 2

y^{^{'}} = \frac{2 x ^{4} e ^{\frac{y}{x}} + x ^{2} y ^{2}}{x ^{3} y}

\int - \frac{sin ( 2 x )}{4} d x

y^{^{''}} + y^{^{'}} + y = cos (t)

\int \frac{7 x ^{6} + 2}{x ^{7} + 2 x} d x

2 x^{2}

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