Science:Math Exam Resources/Courses/MATH100/December 2019/Question 3 (a)/Solution 1

Denote the limit by L . Then,

$L=\lim _{x\rightarrow -\infty }{\frac {5x+1}{x+{\sqrt {x^{2}(1+3/x+2/x^{2})}}}}=\lim _{x\rightarrow -\infty }{\frac {5x+1}{x+|x|{\sqrt {1+3/x+2/x^{2}}}}}.$ Since x is negative, |x|= -x , thus $L=\lim _{x\rightarrow -\infty }{\frac {5x+1}{x-x{\sqrt {1+3/x+2/x^{2}}}}}=\lim _{x\rightarrow -\infty }{\frac {5+1/x}{1-{\sqrt {1+3/x+2/x^{2}}}}}.$

The limit of the numerator is 5, while the denominator is converging to 0 from above, because for very negative values of x we have 1 + 3/x + 2/x^2 < 1. Therefore, the limit diverges to positive infinity.