limit as x approaches 7 of ((1/x-1/7))/(x-7)

1) Compute the limit of

\frac{\frac{1}{x} - \frac{1}{7}}{x - 7}

x \to 7

The limit of

\frac{1}{x}

\frac{1}{7}

when

x \to 7

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

The limit of

\frac{1}{x} - \frac{1}{7}

\frac{1}{7} - \frac{1}{7}

when

x \to 7

⇥1.1) Let's calculate:

\frac{1}{7} - \frac{1}{7}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{7} - \frac{1}{7}

\frac{1 - 1}{7}

⇥

\frac{1 - 1}{7}

⇥ⓘ The numerator and the denominator are computed before the division:

1 - 1

0

⇥

\frac{0}{7}

⇥

0

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

The limit of

x - 7

7 - 7

when

x \to 7

⇥1.2) Let's calculate:

7 - 7

⇥ⓘ Start from

7

and move back

7

on the number line, down to zero:

7 - 7

0

⇥

0

For

\frac{\frac{1}{x} - \frac{1}{7}}{x - 7}

, the limit takes the indeterminate form 0/0 when

x

→

7

2) To find the limit, we factorize the numerator and the denominator:

3) Let's add fractions:

\frac{1}{x} - \frac{1}{7}

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{1}{x} - \frac{1}{7}

\frac{7 - 1 \cdot x}{x \cdot 7}

\frac{7 - 1 \cdot x}{x \cdot 7}

ⓘ Multiplying by 1 changes nothing:

1 \cdot x

x

\frac{7 - x}{x \cdot 7}

We factorize the numerator and the denominator by looking for their roots; those of the denominator will be forbidden values

The roots of the denominator are the forbidden values:

x \neq 0

Factors found:

\frac{7 - x}{7 x}

4) Let's simplify:

\frac{\frac{7 - x}{7 x}}{x - 7}

ⓘ Dividing a fraction by

x - 7

is multiplying its denominator by

x - 7

\frac{\frac{7 - x}{7 x}}{x - 7}

\frac{7 - x}{7 x \cdot (x - 7)}

\frac{7 - x}{7 x \cdot (x - 7)}

We simplify the fraction:

\frac{7 - x}{x - 7}

by the common factor:

- 7 + x

to get the new expression:

-1

- \frac{1}{7 x}

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

The limit of

7 x

7 \cdot 7

when

x \to 7

Let's calculate:

7 \cdot 7

ⅰ

7 \cdot 7

49

49

The limit of

- \frac{1}{7 x}

- \frac{1}{49}

when

x \to 7

So the limit of

\frac{\frac{1}{x} - \frac{1}{7}}{x - 7}

- \frac{1}{49}

when

x \to 7

▶Limit:

- \frac{1}{49}

5) The numeric value of:

- \frac{1}{49}

is:

▷

\approx -2.0408 \times 10^{- 2}