Calculus Examples

$lim_{x \to 7} (x + \sqrt{x - 6}) (x^{2 - 2 x + 1})$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{x \to 7} x + \sqrt{x - 6} \cdot lim_{x \to 7} x^{2 - 2 x + 1}$

Step 1.2

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

$(lim_{x \to 7} x + lim_{x \to 7} \sqrt{x - 6}) \cdot lim_{x \to 7} x^{2 - 2 x + 1}$

Step 1.3

Move the limit under the radical sign.

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 6}) \cdot lim_{x \to 7} x^{2 - 2 x + 1}$

Step 1.4

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

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - lim_{x \to 7} 6}) \cdot lim_{x \to 7} x^{2 - 2 x + 1}$

Step 1.5

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

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot lim_{x \to 7} x^{2 - 2 x + 1}$

Step 1.6

Add $2$ and $1$ .

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot lim_{x \to 7} x^{- 2 x + 3}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $x^{- 2 x + 3}$ as $e^{\ln (x^{- 2 x + 3})}$ .

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot lim_{x \to 7} e^{\ln (x^{- 2 x + 3})}$

Step 2.2

Expand $\ln (x^{- 2 x + 3})$ by moving $- 2 x + 3$ outside the logarithm.

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot lim_{x \to 7} e^{(- 2 x + 3) \ln (x)}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot e^{lim_{x \to 7} (- 2 x + 3) \ln (x)}$

Step 3.2

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

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot e^{lim_{x \to 7} - 2 x + 3 \cdot lim_{x \to 7} \ln (x)}$

Step 3.3

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

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot e^{(- lim_{x \to 7} 2 x + lim_{x \to 7} 3) \cdot lim_{x \to 7} \ln (x)}$

Step 3.4

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

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot e^{(- 2 lim_{x \to 7} x + lim_{x \to 7} 3) \cdot lim_{x \to 7} \ln (x)}$

Step 3.5

Evaluate the limit of $3$ which is constant as $x$ approaches $7$ .

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot e^{(- 2 lim_{x \to 7} x + 3) \cdot lim_{x \to 7} \ln (x)}$

Step 3.6

Move the limit inside the logarithm.

$(lim_{x \to 7} x + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot e^{(- 2 lim_{x \to 7} x + 3) \cdot \ln (lim_{x \to 7} x)}$

Step 4

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

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Step 4.1

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

$(7 + \sqrt{lim_{x \to 7} x - 1 \cdot 6}) \cdot e^{(- 2 lim_{x \to 7} x + 3) \cdot \ln (lim_{x \to 7} x)}$

Step 4.2

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

$(7 + \sqrt{7 - 1 \cdot 6}) \cdot e^{(- 2 lim_{x \to 7} x + 3) \cdot \ln (lim_{x \to 7} x)}$

Step 4.3

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

$(7 + \sqrt{7 - 1 \cdot 6}) \cdot e^{(- 2 \cdot 7 + 3) \cdot \ln (lim_{x \to 7} x)}$

Step 4.4

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

$(7 + \sqrt{7 - 1 \cdot 6}) \cdot e^{(- 2 \cdot 7 + 3) \cdot \ln (7)}$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $- 1$ by $6$ .

$(7 + \sqrt{7 - 6}) \cdot e^{(- 2 \cdot 7 + 3) \cdot \ln (7)}$

Step 5.1.2

Subtract $6$ from $7$ .

$(7 + \sqrt{1}) \cdot e^{(- 2 \cdot 7 + 3) \cdot \ln (7)}$

Step 5.1.3

Any root of $1$ is $1$ .

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

Step 5.2

Add $7$ and $1$ .

$8 \cdot e^{(- 2 \cdot 7 + 3) \cdot \ln (7)}$

Step 5.3

Multiply $- 2$ by $7$ .

$8 \cdot e^{(- 14 + 3) \cdot \ln (7)}$

Step 5.4

Add $- 14$ and $3$ .

$8 e^{- 11 \ln (7)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$8 e^{- 11 \ln (7)}$

Decimal Form:

$4.04586648 \cdot 10^{- 9}$