What is the limit as x approaches infinity of xe^x-x ?

The limit as x approaches infinity of xe^x-x is infinity

limit as x approaches infinity of xe^x-x

x \to \infty lim (x e^{x} - x)

\infty

Solution steps

x \to \infty lim (x e^{x} - x)

Factor out

x e^{x}

from

x e^{x} - x : x e^{x} (1 - \frac{x}{x e ^{x}})

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

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

With the exception of indeterminate form

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

x \to \infty lim (x) = \infty

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

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

= \infty \cdot \infty \cdot 1

Apply Infinity Property:

\infty \infty 1 = \infty

= \infty

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

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

x \to 0 + lim (8 x ln (x))

x \to 0 + lim (1 - 4 x)

x \to 8 + lim (4 x - 1)

What is the limit as x approaches infinity of xe^x-x ?
The limit as x approaches infinity of xe^x-x is infinity