Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Move the limit inside the logarithm.

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

Step 10

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

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

Step 11

Evaluate the limit of $3$ which is constant as $x$ approaches $- 2$ .

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

Step 12

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

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

Step 13

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

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

$\frac{3 \cos (- 2 + 2) - 2}{4 \ln (- 3 - 2 \cdot - 2) + {(- 2)}^{3}}$

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Add $- 2$ and $2$ .

$\frac{3 \cos (0) - 2}{4 \ln (- 3 - 2 \cdot - 2) + {(- 2)}^{3}}$

Step 15.1.2

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

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

Step 15.1.3

Multiply $3$ by $1$ .

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

Step 15.1.4

Subtract $2$ from $3$ .

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

Step 15.2

Simplify the denominator.

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Step 15.2.1

Multiply $- 2$ by $- 2$ .

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

Step 15.2.2

Add $- 3$ and $4$ .

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

Step 15.2.3

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

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

Step 15.2.4

Multiply $4$ by $0$ .

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

Step 15.2.5

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

$\frac{1}{0 - 8}$

Step 15.2.6

Subtract $8$ from $0$ .

$\frac{1}{- 8}$

Step 15.3

Move the negative in front of the fraction.

$- \frac{1}{8}$