MATH100 December 2018
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Question 02 (a)

Compute the following limits:
$\lim _{x_{\rightarrow }\infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}x^{2}}{5x+3}}$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

Multiply both numerator and denominator with ${\sqrt {x^{4}+x^{3}+2}}+x^{2}$

Hint 2

After simplifying the expression, divide by the highest power of $x$ to see which terms will go to zero as $x\to \infty$.

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Solution

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Following the hint, we have
${\begin{aligned}&\lim _{x\to \infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}x^{2}}{5x+3}}\\&=\lim _{x\to \infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}x^{2}}{5x+3}}\cdot {\frac {{\sqrt {x^{4}+x^{3}+2}}+x^{2}}{{\sqrt {x^{4}+x^{3}+2}}+x^{2}}}\\&=\lim _{x\to \infty }{\frac {x^{4}+x^{3}+2x^{4}}{(5x+3)\cdot ({\sqrt {x^{4}+x^{3}+2}}+x^{2})}}\\&=\lim _{x\to \infty }{\frac {x^{3}+2}{(5x+3)({\sqrt {x^{4}+x^{3}+2}}+x^{2})}}.\end{aligned}}$
Now if we divide both the denominator and the numerator by $x^{3}$, we get
${\begin{aligned}&\lim _{x\to \infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}x^{2}}{5x+3}}\\&=\lim _{x\to \infty }{\frac {\frac {x^{3}+2}{x^{3}}}{{\frac {5x+3}{x}}\cdot {\frac {{\sqrt {x^{4}+x^{3}+2}}+x^{2}}{x^{2}}}}}\\&=\lim _{x\to \infty }{\frac {1+{\frac {2}{x^{3}}}}{\left(5+{\frac {3}{x}}\right)\cdot \left({\sqrt {\frac {x^{4}+x^{3}+2}{x^{4}}}}+{\frac {x^{2}}{x^{2}}}\right)}}\\&=\lim _{x\to \infty }{\frac {1+{\frac {2}{x^{3}}}}{\left(5+{\frac {3}{x}}\right)\cdot \left({\sqrt {1+{\frac {1}{x}}+{\frac {2}{x^{4}}}}}+1\right)}}\\&={\frac {1}{5\cdot (1+1)}}\\&={\frac {1}{10}},\end{aligned}}$
so $\color {blue}\lim _{x\to \infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}x^{2}}{5x+3}}={\frac {1}{10}}$.

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