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Exercise

$\lim_{x\to-\infty}\left(\frac{2x^2+3x}{x^2+6}\right)$

Step-by-step Solution

1

As it's an indeterminate limit of type $\frac{\infty}{\infty}$, divide both numerator and denominator by the term of the denominator that tends more quickly to infinity (the term that, evaluated at a large value, approaches infinity faster). In this case, that term is

$\lim_{x\to{- \infty }}\left(\frac{\frac{2x^2+3x}{x^2}}{\frac{x^2+6}{x^2}}\right)$
2

Separate the terms of both fractions

$\lim_{x\to{- \infty }}\left(\frac{\frac{2x^2}{x^2}+\frac{3x}{x^2}}{\frac{x^2}{x^2}+\frac{6}{x^2}}\right)$
3

Simplify the fraction

$\lim_{x\to{- \infty }}\left(\frac{2+\frac{3x}{x^2}}{1+\frac{6}{x^2}}\right)$
4

Simplify the fraction by $x$

$\lim_{x\to{- \infty }}\left(\frac{2+\frac{3}{x}}{1+\frac{6}{x^2}}\right)$
5

Evaluate the limit $\lim_{x\to{- \infty }}\left(\frac{2+\frac{3}{x}}{1+\frac{6}{x^2}}\right)$ by replacing all occurrences of $x$ by $- \infty $

$2$

Final answer to the exercise

$2$

Try other ways to solve this exercise

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  • Solve using L'Hôpital's rule
  • Solve without using l'Hôpital
  • Solve using limit properties
  • Solve using direct substitution
  • Solve the limit using factorization
  • Solve the limit using rationalization
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
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