What is the limit as x approaches 2+of 1/(x-2)-1/(sqrt(x-2)) ?

The limit as x approaches 2+of 1/(x-2)-1/(sqrt(x-2)) is infinity

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limit as x approaches 2+of 1/(x-2)-1/(sqrt(x-2))

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

\infty

Solution steps

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

\frac{1}{x - 2} - \frac{1}{x - 2} = \frac{1}{x - 2} (1 - \frac{\frac{1}{x - 2}}{\frac{1}{x - 2}})

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

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

With the exception of indeterminate form

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

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

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

= \infty \cdot 1

Apply Infinity Property:

c \cdot \infty = \infty

= \infty

What is the limit as x approaches 2+of 1/(x-2)-1/(sqrt(x-2)) ?
The limit as x approaches 2+of 1/(x-2)-1/(sqrt(x-2)) is infinity