limit as x approaches 8 of (sqrt(7+\sqrt[3]{x})-3)/(x-8)

1) Let's evaluate the limit of

\frac{\sqrt{7 + \sqrt[3]{x}} - 3}{x - 8}

when

x \to 8

ⓘ The power with a constant exponent is continuous here: substitute the limit of the base

The limit of

\sqrt[3]{x}

\sqrt[3]{8}

when

x \to 8

⇥1.1) Let's evaluate:

\sqrt[3]{8}

⇥ⓘ The root of order

3

is the number whose power

3

equals the radicand:

\sqrt[3]{8}

2

;

2^{3}

8

⇥

2

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

7 + \sqrt[3]{x}

7 + 2

when

x \to 8

⇥1.2) Let's evaluate:

7 + 2

⇥ⓘ Start from

7

and move forward

2

on the number line:

7 + 2

9

⇥

9

ⓘ The function sqrt is continuous, its limit is its value at the limit of its argument

The limit of

\sqrt{7 + \sqrt[3]{x}}

\sqrt{9}

when

x \to 8

⇥1.3) Let's evaluate:

\sqrt{9}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{9}

3

;

3^{2}

9

⇥

3

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

\sqrt{7 + \sqrt[3]{x}} - 3

3 - 3

when

x \to 8

⇥1.4) Let's calculate:

3 - 3

⇥ⓘ Start from

3

and move back

3

on the number line, down to zero:

3 - 3

0

⇥

0

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

x - 8

8 - 8

when

x \to 8

⇥1.5) Let's calculate:

8 - 8

⇥ⓘ Start from

8

and move back

8

on the number line, down to zero:

8 - 8

0

⇥

0

For

\frac{\sqrt{7 + \sqrt[3]{x}} - 3}{x - 8}

, we encounter the indeterminate form 0/0 as

x

tends to

8

2) We can use L'Hôpital's rule (limit of a/b = limit of a'/b') by computing the derivatives of the numerator and the denominator:

3) Let's differentiate:

\frac{d}{d x} (\sqrt{7 + \sqrt[3]{x}} - 3)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (\sqrt{7 + \sqrt[3]{x}} - 3)

\frac{d}{d x} (\sqrt{7 + \sqrt[3]{x}}) + 0

\frac{d}{d x} (\sqrt{7 + \sqrt[3]{x}}) + 0

We differentiate using:

\frac{d}{d x} (\sqrt{f}) = \frac{d}{d x} (f) \cdot \frac{1}{2 \sqrt{f}}

(where f =

7 + \sqrt[3]{x}

)

ⓘ Adding 0 changes nothing:

\frac{d}{d x} (\sqrt{7 + \sqrt[3]{x}}) + 0

\frac{d}{d x} (\sqrt{7 + \sqrt[3]{x}})

■The derivative of sqrt(x) is 1/(2*sqrt(x)):

\frac{d}{d x} (\sqrt{7 + \sqrt[3]{x}})

\frac{d}{d x} (7 + \sqrt[3]{x}) \cdot \frac{1}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

\frac{d}{d x} (7 + \sqrt[3]{x}) \cdot \frac{1}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{d}{d x} (7 + \sqrt[3]{x}) \cdot \frac{1}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

\frac{\frac{d}{d x} (7 + \sqrt[3]{x}) \cdot 1}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

\frac{\frac{d}{d x} (7 + \sqrt[3]{x})}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (7 + \sqrt[3]{x})

0 + \frac{d}{d x} (\sqrt[3]{x})

\frac{0 + \frac{d}{d x} (\sqrt[3]{x})}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d x} (\sqrt[3]{x})

\frac{d}{d x} (\sqrt[3]{x})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (\sqrt[3]{x})

\frac{1}{3} \cdot x^{- \frac{2}{3}}

\frac{\frac{1}{3} \cdot x^{- \frac{2}{3}}}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{3} \cdot x^{- \frac{2}{3}}

\frac{1 \cdot x^{- \frac{2}{3}}}{3}

\frac{x^{- \frac{2}{3}}}{3}

■The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

x^{- \frac{2}{3}}

\frac{1}{\sqrt[3]{x^{2}}}

\frac{\frac{\frac{1}{\sqrt[3]{x^{2}}}}{3}}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

ⓘ Dividing a fraction by

3

is multiplying its denominator by

3

\frac{\frac{1}{\sqrt[3]{x^{2}}}}{3}

\frac{1}{3 \sqrt[3]{x^{2}}}

■Dividing a fraction by

2 \cdot \sqrt{7 + \sqrt[3]{x}}

is multiplying its denominator by

2 \cdot \sqrt{7 + \sqrt[3]{x}}

\frac{\frac{1}{3 \sqrt[3]{x^{2}}}}{2 \cdot \sqrt{7 + \sqrt[3]{x}}}

\frac{1}{3 \cdot 2 \sqrt[3]{x^{2}} \cdot \sqrt{7 + \sqrt[3]{x}}}

\frac{1}{3 \cdot 2 \sqrt[3]{x^{2}} \cdot \sqrt{7 + \sqrt[3]{x}}}

ⅰ

3 \cdot 2

6

\frac{1}{6 \sqrt[3]{x^{2}} \cdot \sqrt{7 + \sqrt[3]{x}}}

4) Let's perform the differentiation:

\frac{d}{d x} (x - 8)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x - 8)

1 + 0

1 + 0

ⓘ Adding 0 changes nothing:

1 + 0

1

1

Using those derivatives, we obtain a new fraction whose limit can be computed:

ⓘ The power with a constant exponent is continuous here: substitute the limit of the base

The limit of

\sqrt[3]{x^{2}}

\sqrt[3]{8^{2}}

when

x \to 8

⇥4.1) Let's calculate:

\sqrt[3]{8^{2}}

⇥ⓘ The root of order n of a power m is the power m of the root of order n; here, the root of order

3

8

2

\sqrt[3]{8^{2}}

2^{2}

⇥

2^{2}

⇥ⓘ An integer power is a repeated product:

2^{2}

2 \cdot 2

4

⇥

4

ⓘ The limit of a product of finite limits is the product of the limits

The limit of

6 \sqrt[3]{x^{2}} \cdot \sqrt{7 + \sqrt[3]{x}}

6 \cdot 4 \cdot 3

when

x \to 8

⇥4.2) Let's evaluate:

6 \cdot 4 \cdot 3

⇥ⅰ

6 \cdot 4

24

⇥

24 \cdot 3

⇥ⅰ

24 \cdot 3

(20 + 4) \cdot 3

20 \cdot 3 + 4 \cdot 3

60 + 12

72

⇥

72

The limit of

\frac{1}{6 \sqrt[3]{x^{2}} \cdot \sqrt{7 + \sqrt[3]{x}}}

\frac{1}{72}

when

x \to 8

So the limit of

\frac{\sqrt{7 + \sqrt[3]{x}} - 3}{x - 8}

\frac{1}{72}

when

x \to 8

▶Limit:

\frac{1}{72}

5) The numeric value of:

\frac{1}{72}

equals:

▷

\approx 1.3889 \times 10^{- 2}