What is the limit as x approaches 0 of 1/x (1/(x+4.25)-1/(4.25)) ?

The limit as x approaches 0 of 1/x (1/(x+4.25)-1/(4.25)) is -0.05536…

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limit as x approaches 0 of 1/x (1/(x+4.25)-1/(4.25))

x \to 0 lim (\frac{1}{x} (\frac{1}{x + 4.25} - \frac{1}{4.25}))

- 0.05536 \dots

Solution steps

x \to 0 lim (\frac{1}{x} (\frac{1}{x + 4.25} - \frac{1}{4.25}))

\frac{1}{x + 4.25} - \frac{1}{4.25} = \frac{1}{x + 4.25} (1 - \frac{\frac{1}{4.25}}{\frac{1}{x + 4.25}})

= x \to 0 lim (\frac{1}{x} \cdot \frac{1}{x + 4.25} (1 - \frac{\frac{1}{4.25}}{\frac{1}{x + 4.25}}))

Simplify

\frac{1}{x} \cdot \frac{1}{x + 4.25} (1 - \frac{\frac{1}{4.25}}{\frac{1}{x + 4.25}}) : - \frac{0.23529 \dots}{x + 4.25}

= x \to 0 lim (- \frac{0.23529 \dots}{x + 4.25})

Plug in the value

x = 0

= - \frac{0.23529 \dots}{0 + 4.25}

Simplify

= - 0.05536 \dots

What is the limit as x approaches 0 of 1/x (1/(x+4.25)-1/(4.25)) ?
The limit as x approaches 0 of 1/x (1/(x+4.25)-1/(4.25)) is -0.05536…