What is the limit as x approaches infinity of 2^x-x ?

The limit as x approaches infinity of 2^x-x is infinity

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limit as x approaches infinity of 2^x-x

x \to \infty lim (2^{x} - x)

\infty

Solution steps

x \to \infty lim (2^{x} - x)

Apply the following algebraic property

: a + b = a (1 + \frac{b}{a})

2^{x} - x = 2^{x} (1 - \frac{x}{2 ^{x}})

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

x \to \infty lim (2^{x}) = \infty

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

= \infty \cdot 1

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

What is the limit as x approaches infinity of 2^x-x ?
The limit as x approaches infinity of 2^x-x is infinity