What is the limit as x approaches infinity of (3x+2e^x)/(4x-e^{3x)} ?

The limit as x approaches infinity of (3x+2e^x)/(4x-e^{3x)} is 0

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limit as x approaches infinity of (3x+2e^x)/(4x-e^{3x)}

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

0

Solution steps

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

Divide by

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

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

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

x \to \infty lim (\frac{3 x}{e ^{3 x}} + \frac{2}{e ^{2 x}}) = 0

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

= \frac{0}{- 1}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

What is the limit as x approaches infinity of (3x+2e^x)/(4x-e^{3x)} ?
The limit as x approaches infinity of (3x+2e^x)/(4x-e^{3x)} is 0