Calculus Examples

$lim_{x \to - \infty} \tan^{-1} (\frac{\sqrt{1 + 4 x^{6}}}{1 - 2 x^{3}})$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{- \sqrt{0 + 4}}{0 - 1 \cdot 2}$

Step 1.2

Simplify the answer.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Add $0$ and $4$ .

$\frac{- \sqrt{4}}{0 - 1 \cdot 2}$

Step 1.2.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{- \sqrt{2^{2}}}{0 - 1 \cdot 2}$

Step 1.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{- 1 \cdot 2}{0 - 1 \cdot 2}$

Step 1.2.2

Simplify the denominator.

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Step 1.2.2.1

Multiply $- 1$ by $2$ .

$\frac{- 1 \cdot 2}{0 - 2}$

Step 1.2.2.2

Subtract $2$ from $0$ .

$\frac{- 1 \cdot 2}{- 2}$

Step 1.2.3

Multiply $- 1$ by $2$ .

$\frac{- 2}{- 2}$

Step 1.2.4

Divide $- 2$ by $- 2$ .

$1$

Step 2

Substitute $t$ for $\frac{\sqrt{1 + 4 x^{6}}}{1 - 2 x^{3}}$ and let $t$ approach $1$ since $lim_{x \to - \infty} \frac{\sqrt{1 + 4 x^{6}}}{1 - 2 x^{3}} = 1$ .

$lim_{t \to 1} \tan^{-1} (t)$

Step 3

Evaluate the limits by plugging in $1$ for all occurrences of $t$ .

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Step 3.1

Evaluate the limit.

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Step 3.1.1

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

$\tan^{-1} (\frac{- \sqrt{0 + 4}}{0 - 1 \cdot 2})$

Step 3.1.2

Simplify the answer.

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Step 3.1.2.1

Simplify the numerator.

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Step 3.1.2.1.1

Add $0$ and $4$ .

$\tan^{-1} (\frac{- \sqrt{4}}{0 - 1 \cdot 2})$

Step 3.1.2.1.2

Rewrite $4$ as $2^{2}$ .

$\tan^{-1} (\frac{- \sqrt{2^{2}}}{0 - 1 \cdot 2})$

Step 3.1.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\tan^{-1} (\frac{- 1 \cdot 2}{0 - 1 \cdot 2})$

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

Multiply $- 1$ by $2$ .

$\tan^{-1} (\frac{- 1 \cdot 2}{0 - 2})$

Step 3.1.2.2.2

Subtract $2$ from $0$ .

$\tan^{-1} (\frac{- 1 \cdot 2}{- 2})$

Step 3.1.2.3

Multiply $- 1$ by $2$ .

$\tan^{-1} (\frac{- 2}{- 2})$

Step 3.1.2.4

Divide $- 2$ by $- 2$ .

$\tan^{-1} (1)$

Step 3.2

The exact value of $\tan^{-1} (1)$ is $\frac{π}{4}$ .

$\frac{π}{4}$