Calculus Examples

$lim_{x \to - 4} 4 (x^{3} - 8 x^{2} + 20 x - 16) \tan (π x)$

Step 1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$4 lim_{x \to - 4} (x^{3} - 8 x^{2} + 20 x - 16) \tan (π x)$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 4$ .

$4 lim_{x \to - 4} x^{3} - 8 x^{2} + 20 x - 16 \cdot lim_{x \to - 4} \tan (π x)$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 4$ .

$4 (lim_{x \to - 4} x^{3} - lim_{x \to - 4} 8 x^{2} + lim_{x \to - 4} 20 x - lim_{x \to - 4} 16) \cdot lim_{x \to - 4} \tan (π x)$

Step 4

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$4 ({(lim_{x \to - 4} x)}^{3} - lim_{x \to - 4} 8 x^{2} + lim_{x \to - 4} 20 x - lim_{x \to - 4} 16) \cdot lim_{x \to - 4} \tan (π x)$

Step 5

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$4 ({(lim_{x \to - 4} x)}^{3} - 8 lim_{x \to - 4} x^{2} + lim_{x \to - 4} 20 x - lim_{x \to - 4} 16) \cdot lim_{x \to - 4} \tan (π x)$

Step 6

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$4 ({(lim_{x \to - 4} x)}^{3} - 8 {(lim_{x \to - 4} x)}^{2} + lim_{x \to - 4} 20 x - lim_{x \to - 4} 16) \cdot lim_{x \to - 4} \tan (π x)$

Step 7

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$4 ({(lim_{x \to - 4} x)}^{3} - 8 {(lim_{x \to - 4} x)}^{2} + 20 lim_{x \to - 4} x - lim_{x \to - 4} 16) \cdot lim_{x \to - 4} \tan (π x)$

Step 8

Evaluate the limit of $16$ which is constant as $x$ approaches $- 4$ .

$4 ({(lim_{x \to - 4} x)}^{3} - 8 {(lim_{x \to - 4} x)}^{2} + 20 lim_{x \to - 4} x - 1 \cdot 16) \cdot lim_{x \to - 4} \tan (π x)$

Step 9

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

$4 ({(lim_{x \to - 4} x)}^{3} - 8 {(lim_{x \to - 4} x)}^{2} + 20 lim_{x \to - 4} x - 1 \cdot 16) \cdot \tan (lim_{x \to - 4} π x)$

Step 10

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

$4 ({(lim_{x \to - 4} x)}^{3} - 8 {(lim_{x \to - 4} x)}^{2} + 20 lim_{x \to - 4} x - 1 \cdot 16) \cdot \tan (π lim_{x \to - 4} x)$

Step 11

Evaluate the limits by plugging in $- 4$ for all occurrences of $x$ .

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Step 11.1

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$4 ({(- 4)}^{3} - 8 {(lim_{x \to - 4} x)}^{2} + 20 lim_{x \to - 4} x - 1 \cdot 16) \cdot \tan (π lim_{x \to - 4} x)$

Step 11.2

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$4 ({(- 4)}^{3} - 8 {(- 4)}^{2} + 20 lim_{x \to - 4} x - 1 \cdot 16) \cdot \tan (π lim_{x \to - 4} x)$

Step 11.3

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$4 ({(- 4)}^{3} - 8 {(- 4)}^{2} + 20 \cdot - 4 - 1 \cdot 16) \cdot \tan (π lim_{x \to - 4} x)$

Step 11.4

Evaluate the limit of $x$ by plugging in $- 4$ for $x$ .

$4 ({(- 4)}^{3} - 8 {(- 4)}^{2} + 20 \cdot - 4 - 1 \cdot 16) \cdot \tan (π \cdot - 4)$

Step 12

Simplify the answer.

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Step 12.1

Simplify each term.

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Step 12.1.1

Raise $- 4$ to the power of $3$ .

$4 (- 64 - 8 {(- 4)}^{2} + 20 \cdot - 4 - 1 \cdot 16) \cdot \tan (π \cdot - 4)$

Step 12.1.2

Raise $- 4$ to the power of $2$ .

$4 (- 64 - 8 \cdot 16 + 20 \cdot - 4 - 1 \cdot 16) \cdot \tan (π \cdot - 4)$

Step 12.1.3

Multiply $- 8$ by $16$ .

$4 (- 64 - 128 + 20 \cdot - 4 - 1 \cdot 16) \cdot \tan (π \cdot - 4)$

Step 12.1.4

Multiply $20$ by $- 4$ .

$4 (- 64 - 128 - 80 - 1 \cdot 16) \cdot \tan (π \cdot - 4)$

Step 12.1.5

Multiply $- 1$ by $16$ .

$4 (- 64 - 128 - 80 - 16) \cdot \tan (π \cdot - 4)$

Step 12.2

Subtract $128$ from $- 64$ .

$4 (- 192 - 80 - 16) \cdot \tan (π \cdot - 4)$

Step 12.3

Subtract $80$ from $- 192$ .

$4 (- 272 - 16) \cdot \tan (π \cdot - 4)$

Step 12.4

Subtract $16$ from $- 272$ .

$4 \cdot - 288 \cdot \tan (π \cdot - 4)$

Step 12.5

Multiply $4$ by $- 288$ .

$- 1152 \cdot \tan (π \cdot - 4)$

Step 12.6

Move $- 4$ to the left of $π$ .

$- 1152 \cdot \tan (- 4 \cdot π)$

Step 12.7

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

$- 1152 \cdot \tan (0)$

Step 12.8

The exact value of $\tan (0)$ is $0$ .

$- 1152 \cdot 0$

Step 12.9

Multiply $- 1152$ by $0$ .

$0$