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Exercise

$\lim_{x\to\infty}\left(\frac{10n^3+1\:}{n^3}\right)$

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

$\lim_{x\to\infty }\left(\frac{\frac{10n^3+1}{n^3}}{\frac{n^3}{n^3}}\right)$
2

Separate the terms of both fractions

$\lim_{x\to\infty }\left(\frac{\frac{10n^3}{n^3}+\frac{1}{n^3}}{\frac{n^3}{n^3}}\right)$
3

Simplify the fraction $\frac{10n^3}{n^3}$ by $n^3$

$\lim_{x\to\infty }\left(10+\frac{1}{n^3}\right)$
4

The limit of a constant is just the constant

$10+\frac{1}{n^3}$

Final answer to the exercise

$10+\frac{1}{n^3}$

Try other ways to solve this exercise

  • Choose an option
  • 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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