lim x->4 [( x-4 / x^(1/2)-2 ) ] Question related to limits . It’s lim x arrow to 4 . Try to answer this hardcor

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lim x->4 [( x-4 / x^(1/2)-2 ) ]

Question related to limits .
It’s lim x arrow to 4 .
Try to answer this hardcore question.

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Verity 1 year 2021-08-30T10:22:21+00:00 1 Answers 6 views 0

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    2021-08-30T10:23:25+00:00

    [tex]\displaystyle \lim_{x \to 4} \dfrac{x-4}{\sqrt{x} -2} = 4[/tex]

    Step-by-step explanation:

    [tex]\displaystyle \lim_{x \to 4} \dfrac{x-4}{\sqrt{x} -2}[/tex]

    A straightforward substitution of [tex]x=4[/tex] will give us an indeterminate solution. So we use L’Hopital’s rule which states that

    [tex]\displaystyle \lim_{x \to c} \dfrac{f(x)}{g(x)} = \lim_{x \to c} \dfrac{f'(x)}{g'(x)} [/tex]

    where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively. We can see that

    [tex]f'(x) = 1\:\:\:\: \text{and} \:\:\:\:g'(x)= \frac{1}{2} \frac{1}{\sqrt{x}}[/tex]

    Therefore,

    [tex]\displaystyle \lim_{x \to 4} \dfrac{x-4}{\sqrt{x} -2} = \lim_{x \to 4} 2\sqrt{x} = 4[/tex]

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