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Exercise

$\lim_{n\to\infty}\left(\frac{n^2-4\cdot n-2}{n^2}\right)$

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

$\lim_{n\to\infty }\left(\frac{\frac{n^2-4n-2}{n^2}}{\frac{n^2}{n^2}}\right)$
2

Separate the terms of both fractions

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

Simplify the fraction

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

Simplify the fraction by $n$

$\lim_{n\to\infty }\left(1+\frac{-4}{n}+\frac{-2}{n^2}\right)$
5

Evaluate the limit $\lim_{n\to\infty }\left(1+\frac{-4}{n}+\frac{-2}{n^2}\right)$ by replacing all occurrences of $n$ by $\infty $

$1$

Final answer to the exercise

$1$

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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Main Topic: Limits by Direct Substitution

Find limits of functions at a specific point by directly plugging the value into the function.

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