What is the limit as x approaches-infinity of e^{-6x}-2e^{-2x}-1 ?

The limit as x approaches-infinity of e^{-6x}-2e^{-2x}-1 is infinity

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limit as x approaches-infinity of e^{-6x}-2e^{-2x}-1

x \to - \infty lim (e^{- 6 x} - 2 e^{- 2 x} - 1)

\infty

Solution steps

x \to - \infty lim (e^{- 6 x} - 2 e^{- 2 x} - 1)

Apply the following algebraic property

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

e^{- 6 x} - 2 e^{- 2 x} - 1 = e^{- 6 x} (1 - 2 e^{4 x} - e^{6 x})

= x \to - \infty lim (e^{- 6 x} (1 - 2 e^{4 x} - e^{6 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^{- 6 x}) \cdot x \to - \infty lim (1 - 2 e^{4 x} - e^{6 x})

x \to - \infty lim (e^{- 6 x}) = \infty

x \to - \infty lim (1 - 2 e^{4 x} - e^{6 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^{-6x}-2e^{-2x}-1 ?
The limit as x approaches-infinity of e^{-6x}-2e^{-2x}-1 is infinity