Evaluate the following limit: `underset( x rarr 1) ( "lim")(sqrt(5x-4)-sqrt(x))/(x-1)`

Options

A. `7`
B. `3/4`
C. `2`
D. `6/8`

Answer

`2`

Explanation

For finding the limit of the given equation when `x` tends to 1

Substituting `x` as 1 , we get an indeterminant form of `frac(0)(0)`

Rationalizing the given equation

`Rightarrow lim_(x rightarrow 1) frac(sqrt(5 x-4)-sqrt(x))(x-1)=Rightarrow lim_(x rightarrow 1) frac((sqrt(5 x-4)-sqrt(x)))((x-1)) frac((sqrt(5 x-4)+sqrt(x)))((sqrt(5 x-4)+sqrt(x)))`

We have the formula `(a+b)(a-b)=a^(2)-b^(2)`

= `Rightarrow lim_(x rightarrow 1) frac((5 x-4-x))((x-1)) frac(1)((sqrt(5 x-4)+sqrt(x)))`

= `Rightarrow lim_(x rightarrow 1) frac(4(x-1))((x-1)) frac(1)((sqrt(5 x-4)+sqrt(x)))`

`Rightarrow lim_(x rightarrow 1) frac(sqrt(5 x-4)-sqrt(x))(x-1)=Rightarrow lim_(x rightarrow 1) frac(4)(1) frac(1)((sqrt(5 x-4)+sqrt(x)))`

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