Calculus Examples

$lim_{x \to 2} e^{x - 2} \cos (π x)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 2

Move the limit into the exponent.

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

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

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

Step 4

Evaluate the limit of $2$ which is constant as $x$ approaches $2$ .

$e^{lim_{x \to 2} x - 1 \cdot 2} \cdot lim_{x \to 2} \cos (π x)$

Step 5

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

$e^{lim_{x \to 2} x - 1 \cdot 2} \cdot \cos (lim_{x \to 2} π x)$

Step 6

Move the term $π$ outside of the limit because it is constant with respect to $x$ .

$e^{lim_{x \to 2} x - 1 \cdot 2} \cdot \cos (π lim_{x \to 2} x)$

Step 7

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$e^{2 - 1 \cdot 2} \cdot \cos (π lim_{x \to 2} x)$

Step 7.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$e^{2 - 1 \cdot 2} \cdot \cos (π \cdot 2)$

Step 8

Simplify the answer.

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Step 8.1

Multiply $- 1$ by $2$ .

$e^{2 - 2} \cdot \cos (π \cdot 2)$

Step 8.2

Subtract $2$ from $2$ .

$e^{0} \cdot \cos (π \cdot 2)$

Step 8.3

Anything raised to $0$ is $1$ .

$1 \cdot \cos (π \cdot 2)$

Step 8.4

Multiply $\cos (π \cdot 2)$ by $1$ .

$\cos (π \cdot 2)$

Step 8.5

Move $2$ to the left of $π$ .

$\cos (2 \cdot π)$

Step 8.6

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cos (0)$

Step 8.7

The exact value of $\cos (0)$ is $1$ .

$1$