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

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

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

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

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

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

Divide by highest denominator power:

\frac{2 - \frac{3}{x}}{3 - \frac{2}{x}}

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

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

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

= \frac{2}{3}

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