Find the limit of a function \(\displaystyle\lim_{{{x}\to{3}}}{\frac{{\sqrt{{{3}{x}}}-{3}}}{{\sqrt{{{2}{x}-{4}}}-\sqrt{{{2}}}}}}\)

Aliana Alvarez

Aliana Alvarez

Answered question

2022-04-12

Find the limit of a function
limx33x32x42

Answer & Explanation

Raina Blackburn

Raina Blackburn

Beginner2022-04-13Added 10 answers

Your error is that after rationalizing the denominator, you cannot just plug in 3 for x, because the denominator still evaluates to 0 (not 2, as you seem to think).
F(x)=32(x3x3x21x3)
So
limx3F(x)=32limx3x3x3limx3x21x3=32g(3)h(3)
where g(x)=x and h(x)=x2
Since g(x)=12x,g(3)=123. Since h(x)12x2, then h(3)=12. So
limx3F(x)=3212312=1222=12

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