Math

QuestionFind the limit as xx approaches 0 for the expression excosxe^{-x} \cos x.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit of the function excosxe^{-x} \cos x as xx approaches 00. . We know that the limit of a product of two functions is the product of their limits, if both limits exist.
3. We know the limit of exe^{-x} as xx approaches 00 and the limit of cosx\cos x as xx approaches 00.

STEP 2

We can express the limit of the function as the product of the limits of exe^{-x} and cosx\cos x.
limx0excosx=limx0ex×limx0cosx\lim{x \rightarrow0} e^{-x} \cos x = \lim{x \rightarrow0} e^{-x} \times \lim{x \rightarrow0} \cos x

STEP 3

We find the limit of exe^{-x} as xx approaches 00.
limx0ex=e0=1\lim{x \rightarrow0} e^{-x} = e^{-0} =1

STEP 4

We find the limit of cosx\cos x as xx approaches 00.
limx0cosx=cos0=1\lim{x \rightarrow0} \cos x = \cos0 =1

STEP 5

Substitute the limits of exe^{-x} and cosx\cos x into the equation.
limx0excosx=1×1=1\lim{x \rightarrow0} e^{-x} \cos x =1 \times1 =1So, the limit of the function excosxe^{-x} \cos x as xx approaches 00 is 11.

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