Solved: "Knowing thatlimx→ax2−a3xax−a=12Find out a."

Sebastian Fox

Answered question

2022-04-16

Knowing that

lim_{x \to a} \frac{x^{2} - \sqrt{a^{3} x}}{\sqrt{a x} - a} = 12

Find out a.

cutimnm135imsa

Beginner2022-04-17Added 21 answers

First note that

a > 0

a < 0

, then

lim_{x \to a} \frac{x^{2} - \sqrt{a^{3} x}}{\sqrt{a x} - a} = \frac{a^{2} - a^{2}}{- a - a} = 0

a = 0

, then

lim_{x \to a} \frac{x^{2} - \sqrt{a^{3} x}}{\sqrt{a x} - a}

doesn't exist.
Hence,

a > 0

Let

\sqrt{a x} = y

i.e.

x = \frac{y^{2}}{a}

. Note that as

x \to a, y \to a

Hence,

lim_{x \to a} \frac{x^{2} - \sqrt{a^{3} x}}{\sqrt{a x} - a} = lim_{y \to a} \frac{y}{a^{2}} \frac{y^{3} - a^{3}}{y - a} = \frac{3 a^{2}}{a} = 3 a

Hence,

a = 4

annieljcddj0

Beginner2022-04-18Added 15 answers

One way is application of L'hoipitals rule. Since both denominator and numerator in

lim_{x \to a} \frac{x^{2} - \sqrt{a^{3} x}}{\sqrt{a x} - a}

go to zero, by l'hopitals rule this is the same as

lim_{x \to a} \frac{2 x - a^{\frac{3}{2}} \frac{1}{2} x^{- \frac{1}{2}}}{\frac{\sqrt{a}}{2} x^{- \frac{1}{2}}}

Simplifying the expression we get that this equals

lim_{x \to a} \frac{4 x^{\frac{3}{2}} - a^{\frac{3}{2}}}{\sqrt{a}} = \frac{3 a^{\frac{3}{2}}}{\sqrt{a}} = 3 a

Thus choose

a = 4

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