Science:Math Exam Resources/Courses/MATH100/December 2018/Question 02 (a)

{{#incat:MER QGQ flag|{{#incat:MER QGH flag|{{#incat:MER QGS flag|}}}}}}

MATH100 December 2018

• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 • Q4 • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q6 • Q7 • Q8 • Q9 • Q10 • Q11 (a) • Q11 (b) •

Other MATH100 Exams

• December 2011 • December 2010 • December 2012 • December 2013 • December 2014 • December 2015 • December 2016 • December 2018 •

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 $x\to -\infty$ .

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution
Following the hint, we have ${\begin{aligned}&\lim _{x\to -\infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}-x^{2}}{5x+3}}\\&=\lim _{x\to -\infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}-x^{2}}{5x+3}}\cdot {\frac {{\sqrt {x^{4}+x^{3}+2}}+x^{2}}{{\sqrt {x^{4}+x^{3}+2}}+x^{2}}}\\&=\lim _{x\to -\infty }{\frac {x^{4}+x^{3}+2-x^{4}}{(5x+3)\cdot ({\sqrt {x^{4}+x^{3}+2}}+x^{2})}}\\&=\lim _{x\to -\infty }{\frac {x^{3}+2}{(5x+3)({\sqrt {x^{4}+x^{3}+2}}+x^{2})}}.\end{aligned}}$ 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 $\color {blue}\lim _{x\to -\infty }{\frac {{\sqrt {x^{4}+x^{3}+2}}-x^{2}}{5x+3}}={\frac {1}{10}}$ .