Question 938194
I'll do part a) to get you started.



*[Tex \LARGE \lim_{x \to \infty} \frac{\sqrt{4x^2-3}}{2x+3}]



*[Tex \LARGE \lim_{x \to \infty} \frac{\sqrt{x^2(4-\frac{3}{x^2})}}{2x+3}] Factor out {{{x^2}}} (the radicand in the numerator)



*[Tex \LARGE \lim_{x \to \infty} \frac{\sqrt{x^2}*\sqrt{4-\frac{3}{x^2}}}{2x+3}] Break up the root



*[Tex \LARGE \lim_{x \to \infty} \frac{x*\sqrt{4-\frac{3}{x^2}}}{2x+3}]



*[Tex \LARGE \lim_{x \to \infty} \frac{\frac{1}{x}*x*\sqrt{4-\frac{3}{x^2}}}{\frac{1}{x}*(2x+3)}] Multiply top and bottom by {{{1/x}}}



*[Tex \LARGE \lim_{x \to \infty} \frac{\sqrt{4-\frac{3}{x^2}}}{2+\frac{3}{x}}]



*[Tex \LARGE \frac{\sqrt{4-0}}{2+0}] Use the idea that *[Tex \Large \lim_{x \to \infty} \left(\frac{1}{x^p}\right) = 0] (p is any positive real number)



*[Tex \LARGE \frac{\sqrt{4}}{2}]



*[Tex \LARGE \frac{2}{2}]



*[Tex \LARGE 1]



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Therefore, *[Tex \LARGE \lim_{x \to \infty} \frac{\sqrt{4x^2-3}}{2x+3} = 1]

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