What is the limit as x approaches infinity of (e^{5x})/(3-e^{2x)} ?

The limit as x approaches infinity of (e^{5x})/(3-e^{2x)} is -infinity

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limit as x approaches infinity of (e^{5x})/(3-e^{2x)}

x \to \infty lim (\frac{e ^{5 x}}{3 - e ^{2 x}})

- \infty

Solution steps

x \to \infty lim (\frac{e ^{5 x}}{3 - e ^{2 x}})

Divide by

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

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

x \to \infty lim (e^{3 x}) = \infty

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

= \frac{\infty}{- 1}

Apply Infinity Property:

\frac{\infty}{- c} = - \infty

= - \infty

x \to 0 lim (x cos (2 x))

x \to 0 lim (sin (x) cos (x))

x \to \infty lim (- 8 (\frac{3}{x}))

x \to 0 lim (\frac{5 ^{x} - 6 ^{x}}{x})

x \to 0 + lim (x ∣ x ∣)

What is the limit as x approaches infinity of (e^{5x})/(3-e^{2x)} ?
The limit as x approaches infinity of (e^{5x})/(3-e^{2x)} is -infinity