Evaluate limit as x approaches infinity of e^(-2x)cos(x)

$lim_{x \to \infty} e^{- 2 x} \cos (x)$

Take the limit of each term.

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Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\infty$ .

$lim_{x \to \infty} e^{- 2 x} \cdot lim_{x \to \infty} \cos (x)$

Since the exponent $- 2 x$ approaches $- \infty$ , the quantity $e^{- 2 x}$ approaches $0$ .

$0 \cdot lim_{x \to \infty} \cos (x)$

Move the limit inside the trig function because cosine is continuous.

$0 \cdot \cos (lim_{x \to \infty} x)$

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$0 \cdot \cos (\infty)$

Multiply $0$ by $\cos (\infty)$ .

$0$

Evaluate limit as x approaches infinity of e^(-2x)cos(x)

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