Calculus Examples

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

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 2

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

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

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

Step 4

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 5

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

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

Step 6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

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

Step 7

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 8

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 9

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

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

Step 10

Move the limit inside the logarithm.

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

Step 11

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

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

Step 12

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

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

Step 13

Evaluate the limit of $4$ which is constant as $x$ approaches $3$ .

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

Step 14

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 14.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 14.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 14.3

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 15

Simplify the answer.

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Step 15.1

Simplify each term.

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Step 15.1.1

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

$7 \cdot 9 - 2 \frac{1}{3 + 1} \cdot \ln (3 - 1 \cdot 2) + 4$

Step 15.1.2

Multiply $7$ by $9$ .

$63 - 2 \frac{1}{3 + 1} \cdot \ln (3 - 1 \cdot 2) + 4$

Step 15.1.3

Add $3$ and $1$ .

$63 - 2 (\frac{1}{4}) \cdot \ln (3 - 1 \cdot 2) + 4$

Step 15.1.4

Cancel the common factor of $2$ .

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Step 15.1.4.1

Factor $2$ out of $- 2$ .

$63 + 2 (- 1) \frac{1}{4} \cdot \ln (3 - 1 \cdot 2) + 4$

Step 15.1.4.2

Factor $2$ out of $4$ .

$63 + 2 \cdot - 1 \frac{1}{2 \cdot 2} \cdot \ln (3 - 1 \cdot 2) + 4$

Step 15.1.4.3

Cancel the common factor.

$63 + 2 \cdot - 1 \frac{1}{2 \cdot 2} \cdot \ln (3 - 1 \cdot 2) + 4$

Step 15.1.4.4

Rewrite the expression.

$63 - \frac{1}{2} \cdot \ln (3 - 1 \cdot 2) + 4$

Step 15.1.5

Multiply $- 1$ by $2$ .

$63 - \frac{1}{2} \cdot \ln (3 - 2) + 4$

Step 15.1.6

Subtract $2$ from $3$ .

$63 - \frac{1}{2} \cdot \ln (1) + 4$

Step 15.1.7

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

$63 - \frac{1}{2} \cdot 0 + 4$

Step 15.1.8

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

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Step 15.1.8.1

Multiply $0$ by $- 1$ .

$63 + 0 (\frac{1}{2}) + 4$

Step 15.1.8.2

Multiply $0$ by $\frac{1}{2}$ .

$63 + 0 + 4$

Step 15.2

Add $63$ and $0$ .

$63 + 4$

Step 15.3

Add $63$ and $4$ .

$67$