What is the limit as x approaches 2+of (|x|)/(2-x) ?

The limit as x approaches 2+of (|x|)/(2-x) is -infinity

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limit as x approaches 2+of (|x|)/(2-x)

x \to 2 + lim (\frac{∣ x ∣}{2 - x})

- \infty

Solution steps

x \to 2 + lim (\frac{∣ x ∣}{2 - x})

x

is positive when

x \to 2 +

. Therefore

∣ x ∣ = x

= x \to 2 + lim (\frac{x}{2 - x})

Divide by highest denominator power:

\frac{1}{\frac{2}{x} - 1}

= x \to 2 + lim (\frac{1}{\frac{2}{x} - 1})

For

x

approaching

2

from the right

, x > 2 \Rightarrow \frac{2}{x} - 1 < 0

The denominator is a negative quantity approaching 0 from the left

= - \infty

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What is the limit as x approaches 2+of (|x|)/(2-x) ?
The limit as x approaches 2+of (|x|)/(2-x) is -infinity