Calculus Examples

$lim_{x \to 4} \frac{3 (\ln (9 x + 10) + 1)}{4 ({(2 x + 2)}^{\frac{1}{3}} + 1)}$

Step 1

Move the term $\frac{3}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{4} lim_{x \to 4} \frac{\ln (9 x + 10) + 1}{{(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 2

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

$\frac{3}{4} \cdot \frac{lim_{x \to 4} \ln (9 x + 10) + 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 3

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

$\frac{3}{4} \cdot \frac{lim_{x \to 4} \ln (9 x + 10) + lim_{x \to 4} 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 4

Move the limit inside the logarithm.

$\frac{3}{4} \cdot \frac{\ln (lim_{x \to 4} 9 x + 10) + lim_{x \to 4} 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 5

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

$\frac{3}{4} \cdot \frac{\ln (lim_{x \to 4} 9 x + lim_{x \to 4} 10) + lim_{x \to 4} 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 6

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + lim_{x \to 4} 10) + lim_{x \to 4} 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 7

Evaluate the limit of $10$ which is constant as $x$ approaches $4$ .

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + lim_{x \to 4} 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + 1}$

Step 9

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

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + 1}{lim_{x \to 4} {(2 x + 2)}^{\frac{1}{3}} + lim_{x \to 4} 1}$

Step 10

Move the exponent $\frac{1}{3}$ from ${(2 x + 2)}^{\frac{1}{3}}$ outside the limit using the Limits Power Rule.

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + 1}{{(lim_{x \to 4} 2 x + 2)}^{\frac{1}{3}} + lim_{x \to 4} 1}$

Step 11

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

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + 1}{{(lim_{x \to 4} 2 x + lim_{x \to 4} 2)}^{\frac{1}{3}} + lim_{x \to 4} 1}$

Step 12

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

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + 1}{{(2 lim_{x \to 4} x + lim_{x \to 4} 2)}^{\frac{1}{3}} + lim_{x \to 4} 1}$

Step 13

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

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + 1}{{(2 lim_{x \to 4} x + 2)}^{\frac{1}{3}} + lim_{x \to 4} 1}$

Step 14

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\frac{3}{4} \cdot \frac{\ln (9 lim_{x \to 4} x + 10) + 1}{{(2 lim_{x \to 4} x + 2)}^{\frac{1}{3}} + 1}$

Step 15

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

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Step 15.1

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

$\frac{3}{4} \cdot \frac{\ln (9 \cdot 4 + 10) + 1}{{(2 lim_{x \to 4} x + 2)}^{\frac{1}{3}} + 1}$

Step 15.2

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

$\frac{3}{4} \cdot \frac{\ln (9 \cdot 4 + 10) + 1}{{(2 \cdot 4 + 2)}^{\frac{1}{3}} + 1}$

Step 16

Simplify the answer.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Multiply $9$ by $4$ .

$\frac{3}{4} \cdot \frac{\ln (36 + 10) + 1}{{(2 \cdot 4 + 2)}^{\frac{1}{3}} + 1}$

Step 16.1.2

Add $36$ and $10$ .

$\frac{3}{4} \cdot \frac{\ln (46) + 1}{{(2 \cdot 4 + 2)}^{\frac{1}{3}} + 1}$

Step 16.2

Simplify the denominator.

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Step 16.2.1

Multiply $2$ by $4$ .

$\frac{3}{4} \cdot \frac{\ln (46) + 1}{{(8 + 2)}^{\frac{1}{3}} + 1}$

Step 16.2.2

Add $8$ and $2$ .

$\frac{3}{4} \cdot \frac{\ln (46) + 1}{10^{\frac{1}{3}} + 1}$

Step 16.3

Multiply $\frac{3}{4}$ by $\frac{\ln (46) + 1}{10^{\frac{1}{3}} + 1}$ .

$\frac{3 (\ln (46) + 1)}{4 (10^{\frac{1}{3}} + 1)}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{3 (\ln (46) + 1)}{4 (10^{\frac{1}{3}} + 1)}$

Decimal Form:

$1.14806024 \dots$