Question

Q.
[tex]\sf nilai \: lim_{x \: \to \: 1} \frac{ \sqrt{5x - 4} - x}{ {x}^{2} - 1}[/tex]
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Answer 1

Limit

[Metode L'Hopital]

..

[tex]\displaystyle\lim_{x \to 1} \frac{ \sqrt{5x - 4} - x}{ {x}^{2} - 1}[/tex]

[tex]\displaystyle\lim_{x \to 1} \frac{ \frac{d}{dx}( \sqrt{5x - 4} - x)}{ \frac{d}{dx} ({x}^{2} - 1)}[/tex]

[tex]\displaystyle\lim_{x \to 1} \frac{ \frac{5}{2 \sqrt{5x - 4} } - 1 }{ 2x}[/tex]

[tex]\displaystyle\lim_{x \to 1} \frac{ \frac{5 - 2 \sqrt{5x - 4} }{2 \sqrt{5x - 4} } }{ 2x}[/tex]

[tex]\displaystyle\lim_{x \to 1} \frac{ {5 - 2 \sqrt{5x - 4} } }{ 4x \sqrt{5x - 4} }[/tex]

..

[tex] = \frac{5 - 2 \sqrt{5(1) - 4} }{4(1) \sqrt{5(1) - 4} } [/tex]

[tex] = \frac{5 - 2}{4 - 1} [/tex]

[tex] = \boxed{ \frac{3}{4} }[/tex]

..

[tex]\begin{array}{lr}\texttt{Selamat Menunaikan Ibadah Puasa}\end{array}[/tex]

[tex]\boxed{\colorbox{ccddff}{Answered by Danial Alf'at | 01 - 04 - 2023}}[/tex]

Answer 2

[tex]\tt nilai \: lim_{x \: \to \: 1} \frac{ \sqrt{5x - 4} - x}{ {x}^{2} - 1}[/tex]

= (x - 4)(-1)/(x + 1)(√5x - 4 + x)

= (1 - 4)(-1)/(1 + 1)(√(5 × 1) - 4 + 1)

= (-3(-1)/2(√5 - 4 + 1)

= (3(1)/2(√1 + 1)

= 3/2(1 + 1)

= 3/(2 × 2)

= 3/4

[tex]\mathcal{ \color{cyan} \:INFINITE \: WORLD}[/tex]