Question

it: \\( \lim _ { x \
ightarrow 1 } \frac { 3 - 3 x } { 1 - \sqrt { x } } \\)

Explanation:

Step1: Factor numerator and rationalize denominator

Factor numerator: \(3 - 3x=3(1 - x)=3(1-\sqrt{x})(1 + \sqrt{x})\).
The original limit becomes \(\lim_{x
ightarrow1}\frac{3(1 - \sqrt{x})(1+\sqrt{x})}{1-\sqrt{x}}\).

Step2: Simplify the expression

Cancel out the common factor \((1 - \sqrt{x})\) (for \(x
eq1\)), we get \(\lim_{x
ightarrow1}3(1+\sqrt{x})\).

Step3: Substitute \(x = 1\)

Substitute \(x = 1\) into \(3(1+\sqrt{x})\), we have \(3(1+\sqrt{1})\).

Answer:

\(6\)