Calculus Examples

$lim_{x \to - 2} \frac{\ln (3 x + 7) - 2 x^{3}}{3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 2$ .

$\frac{lim_{x \to - 2} \ln (3 x + 7) - 2 x^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 2

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

$\frac{lim_{x \to - 2} \ln (3 x + 7) - lim_{x \to - 2} 2 x^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 3

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to - 2} 3 x + 7) - lim_{x \to - 2} 2 x^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 4

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

$\frac{\ln (lim_{x \to - 2} 3 x + lim_{x \to - 2} 7) - lim_{x \to - 2} 2 x^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 5

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

$\frac{\ln (3 lim_{x \to - 2} x + lim_{x \to - 2} 7) - lim_{x \to - 2} 2 x^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 6

Evaluate the limit of $7$ which is constant as $x$ approaches $- 2$ .

$\frac{\ln (3 lim_{x \to - 2} x + 7) - lim_{x \to - 2} 2 x^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 7

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 lim_{x \to - 2} x^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 8

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - 3 x^{3}}$

Step 9

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{lim_{x \to - 2} 3 \tan (- 4 - 2 x) - lim_{x \to - 2} 3 x^{3}}$

Step 10

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 lim_{x \to - 2} \tan (- 4 - 2 x) - lim_{x \to - 2} 3 x^{3}}$

Step 11

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 \tan (lim_{x \to - 2} - 4 - 2 x) - lim_{x \to - 2} 3 x^{3}}$

Step 12

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 \tan (- lim_{x \to - 2} 4 - lim_{x \to - 2} 2 x) - lim_{x \to - 2} 3 x^{3}}$

Step 13

Evaluate the limit of $4$ which is constant as $x$ approaches $- 2$ .

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 \tan (- 1 \cdot 4 - lim_{x \to - 2} 2 x) - lim_{x \to - 2} 3 x^{3}}$

Step 14

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 lim_{x \to - 2} x) - lim_{x \to - 2} 3 x^{3}}$

Step 15

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 lim_{x \to - 2} x) - 3 lim_{x \to - 2} x^{3}}$

Step 16

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

$\frac{\ln (3 lim_{x \to - 2} x + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 lim_{x \to - 2} x) - 3 {(lim_{x \to - 2} x)}^{3}}$

Step 17

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

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Step 17.1

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

$\frac{\ln (3 \cdot - 2 + 7) - 2 {(lim_{x \to - 2} x)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 lim_{x \to - 2} x) - 3 {(lim_{x \to - 2} x)}^{3}}$

Step 17.2

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

$\frac{\ln (3 \cdot - 2 + 7) - 2 {(- 2)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 lim_{x \to - 2} x) - 3 {(lim_{x \to - 2} x)}^{3}}$

Step 17.3

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

$\frac{\ln (3 \cdot - 2 + 7) - 2 {(- 2)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(lim_{x \to - 2} x)}^{3}}$

Step 17.4

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

$\frac{\ln (3 \cdot - 2 + 7) - 2 {(- 2)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18

Simplify the answer.

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Step 18.1

Simplify the numerator.

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Step 18.1.1

Multiply $3$ by $- 2$ .

$\frac{\ln (- 6 + 7) - 2 {(- 2)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.1.2

Add $- 6$ and $7$ .

$\frac{\ln (1) - 2 {(- 2)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.1.3

The natural logarithm of $1$ is $0$ .

$\frac{0 - 2 {(- 2)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.1.4

Multiply $- 2$ by ${(- 2)}^{3}$ by adding the exponents.

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Step 18.1.4.1

Multiply $- 2$ by ${(- 2)}^{3}$ .

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Step 18.1.4.1.1

Raise $- 2$ to the power of $1$ .

$\frac{0 + {(- 2)}^{1} {(- 2)}^{3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.1.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{0 + {(- 2)}^{1 + 3}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.1.4.2

Add $1$ and $3$ .

$\frac{0 + {(- 2)}^{4}}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.1.5

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

$\frac{0 + 16}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.1.6

Add $0$ and $16$ .

$\frac{16}{3 \tan (- 1 \cdot 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.2

Simplify the denominator.

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Step 18.2.1

Simplify each term.

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Step 18.2.1.1

Multiply $- 1$ by $4$ .

$\frac{16}{3 \tan (- 4 - 1 \cdot 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.2.1.2

Multiply $- 1 \cdot 2 \cdot - 2$ .

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Step 18.2.1.2.1

Multiply $- 1$ by $2$ .

$\frac{16}{3 \tan (- 4 - 2 \cdot - 2) - 3 {(- 2)}^{3}}$

Step 18.2.1.2.2

Multiply $- 2$ by $- 2$ .

$\frac{16}{3 \tan (- 4 + 4) - 3 {(- 2)}^{3}}$

Step 18.2.2

Add $- 4$ and $4$ .

$\frac{16}{3 \tan (0) - 3 {(- 2)}^{3}}$

Step 18.2.3

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

$\frac{16}{3 \cdot 0 - 3 {(- 2)}^{3}}$

Step 18.2.4

Multiply $3$ by $0$ .

$\frac{16}{0 - 3 {(- 2)}^{3}}$

Step 18.2.5

Raise $- 2$ to the power of $3$ .

$\frac{16}{0 - 3 \cdot - 8}$

Step 18.2.6

Multiply $- 3$ by $- 8$ .

$\frac{16}{0 + 24}$

Step 18.2.7

Add $0$ and $24$ .

$\frac{16}{24}$

Step 18.3

Cancel the common factor of $16$ and $24$ .

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Step 18.3.1

Factor $8$ out of $16$ .

$\frac{8 (2)}{24}$

Step 18.3.2

Cancel the common factors.

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Step 18.3.2.1

Factor $8$ out of $24$ .

$\frac{8 \cdot 2}{8 \cdot 3}$

Step 18.3.2.2

Cancel the common factor.

$\frac{8 \cdot 2}{8 \cdot 3}$

Step 18.3.2.3

Rewrite the expression.

$\frac{2}{3}$