"Find the limits: limx→44x−x22−x" was answered by students like you

geduiwelh

Answered question

2020-12-16

Find the limits:

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

estenutC

Skilled2020-12-17Added 81 answers

We have to find the limits:

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

It is

\frac{0}{0}

form
Simplifying the limit by using rationalization of denominator.
Multiplying and dividing by

(2 + \sqrt{x}),

lim_{x \to 4} \frac{4 x - x^{2}}{2 - \sqrt{x}} \times \frac{(2 + \sqrt{x})}{(2 + \sqrt{x})}

We know the idenity,

(a - b) (a + b) = a^{2} - b^{2}

Therefore,

lim_{x \to 4} \frac{4 x - x^{2}}{2 - \sqrt{x}} \times \frac{(2 + \sqrt{x})}{(2 + \sqrt{x})} = lim_{x \to 4} \frac{(4 x - x^{2}) (2 + \sqrt{x})}{(2)^{2} - (\sqrt{x})^{2}}

lim_{x \to 4} \frac{x (4 - x) (2 + \sqrt{x})}{4 - x} (taken x common from 4 x - x^{2})

lim_{x \to 4} x (2 + \sqrt{x})

= 4 (2 + \sqrt{4})

= 4 (2 + 2)

= 4 \times 4

= 16

Hence, value of limit is 16.

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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