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

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

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limit as x approaches infinity of e^x-x

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

\infty

Solution steps

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

Factor out

e^{x}

from

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

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

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

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

= \infty \cdot 1

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

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