Calculus Examples

$lim_{x \to 7} \ln (\frac{x}{\sqrt{x - 6}})$

Step 1

Move the limit inside the logarithm.

$\ln (lim_{x \to 7} \frac{x}{\sqrt{x - 6}})$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $7$ .

$\ln (\frac{lim_{x \to 7} x}{lim_{x \to 7} \sqrt{x - 6}})$

Step 3

Move the limit under the radical sign.

$\ln (\frac{lim_{x \to 7} x}{\sqrt{lim_{x \to 7} x - 6}})$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$\ln (\frac{lim_{x \to 7} x}{\sqrt{lim_{x \to 7} x - lim_{x \to 7} 6}})$

Step 5

Evaluate the limit of $6$ which is constant as $x$ approaches $7$ .

$\ln (\frac{lim_{x \to 7} x}{\sqrt{lim_{x \to 7} x - 1 \cdot 6}})$

Step 6

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\ln (\frac{7}{\sqrt{lim_{x \to 7} x - 1 \cdot 6}})$

Step 6.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$\ln (\frac{7}{\sqrt{7 - 1 \cdot 6}})$

Step 7

Simplify the answer.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Multiply $- 1$ by $6$ .

$\ln (\frac{7}{\sqrt{7 - 6}})$

Step 7.1.2

Subtract $6$ from $7$ .

$\ln (\frac{7}{\sqrt{1}})$

Step 7.1.3

Any root of $1$ is $1$ .

$\ln (\frac{7}{1})$

Step 7.2

Divide $7$ by $1$ .

$\ln (7)$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\ln (7)$

Decimal Form:

$1.94591014 \dots$