Evaluate lim \: ( (1)/( √(x) ) - 1)/( √(x) - 1) \: as \: x \: approaches \: 1 ​

Question

Evaluate


`lim \: ( (1)/( √(x) ) - 1)/( √(x) - 1) \: as \: x \: approaches \: 1`
​

Answer 1

-1

Explanation:

In many cases, the simplified expression is not undefined at the point of interest.

`(\left((1)/(√(x))-1\right))/(√(x)-1)=(\left((1-√(x))/(√(x))\right))/(√(x)-1)=(-1)/(√(x))`

This can be evaluated at x=1:

-1/√1 = -1

Then, the limit is ...

`\boxed{\lim\limits_(x\to 1)(\left((1)/(√(x))-1\right))/(√(x)-1)=-1}`

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A graph confirms this conclusion.