Calculus Examples

$lim_{x \to 4} 2 x^{3 \sqrt{x^{2} + 7}}$

Step 1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 4} x^{3 \sqrt{x^{2} + 7}}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $x^{3 \sqrt{x^{2} + 7}}$ as $e^{\ln (x^{3 \sqrt{x^{2} + 7}})}$ .

$2 lim_{x \to 4} e^{\ln (x^{3 \sqrt{x^{2} + 7}})}$

Step 2.2

Expand $\ln (x^{3 \sqrt{x^{2} + 7}})$ by moving $3 \sqrt{x^{2} + 7}$ outside the logarithm.

$2 lim_{x \to 4} e^{3 \sqrt{x^{2} + 7} \ln (x)}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$2 e^{lim_{x \to 4} 3 \sqrt{x^{2} + 7} \ln (x)}$

Step 3.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$2 e^{3 lim_{x \to 4} \sqrt{x^{2} + 7} \ln (x)}$

Step 3.3

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

$2 e^{3 lim_{x \to 4} \sqrt{x^{2} + 7} \cdot lim_{x \to 4} \ln (x)}$

Step 3.4

Move the limit under the radical sign.

$2 e^{3 \sqrt{lim_{x \to 4} x^{2} + 7} \cdot lim_{x \to 4} \ln (x)}$

Step 3.5

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

$2 e^{3 \sqrt{lim_{x \to 4} x^{2} + lim_{x \to 4} 7} \cdot lim_{x \to 4} \ln (x)}$

Step 3.6

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

$2 e^{3 \sqrt{{(lim_{x \to 4} x)}^{2} + lim_{x \to 4} 7} \cdot lim_{x \to 4} \ln (x)}$

Step 3.7

Evaluate the limit of $7$ which is constant as $x$ approaches $4$ .

$2 e^{3 \sqrt{{(lim_{x \to 4} x)}^{2} + 7} \cdot lim_{x \to 4} \ln (x)}$

Step 3.8

Move the limit inside the logarithm.

$2 e^{3 \sqrt{{(lim_{x \to 4} x)}^{2} + 7} \cdot \ln (lim_{x \to 4} x)}$

Step 4

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

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Step 4.1

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

$2 e^{3 \sqrt{4^{2} + 7} \cdot \ln (lim_{x \to 4} x)}$

Step 4.2

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

$2 e^{3 \sqrt{4^{2} + 7} \cdot \ln (4)}$

Step 5

Simplify the answer.

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Step 5.1

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

$2 e^{3 \sqrt{16 + 7} \cdot \ln (4)}$

Step 5.2

Add $16$ and $7$ .

$2 e^{3 \sqrt{23} \ln (4)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 e^{3 \sqrt{23} \ln (4)}$

Decimal Form:

$918681247.931872$