Evaluate: \lim_{n arrow+infinity} (((n+1) \times (n+2) \times (n+3))/(n^3))

Expression: $\lim_{n \rightarrow +\infty} \left(\frac{ \left( n+1 \right) \times \left( n+2 \right) \times \left( n+3 \right) }{ {n}^{3} }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{n \rightarrow +\infty} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( \left( n+1 \right) \times \left( n+2 \right) \times \left( n+3 \right) \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( {n}^{3} \right) }\right)$

Find the derivative

$\lim_{n \rightarrow +\infty} \left(\frac{ 3{n}^{2}+12n+11 }{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( {n}^{3} \right) }\right)$

Find the derivative

$\lim_{n \rightarrow +\infty} \left(\frac{ 3{n}^{2}+12n+11 }{ 3{n}^{2} }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{n \rightarrow +\infty} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( 3{n}^{2}+12n+11 \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( 3{n}^{2} \right) }\right)$

Find the derivative

$\lim_{n \rightarrow +\infty} \left(\frac{ 6n+12 }{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( 3{n}^{2} \right) }\right)$

Find the derivative

$\lim_{n \rightarrow +\infty} \left(\frac{ 6n+12 }{ 6n }\right)$

Factor out $6$ from the expression

$\lim_{n \rightarrow +\infty} \left(\frac{ 6\left( n+2 \right) }{ 6n }\right)$

Cancel out the common factor $6$

$\lim_{n \rightarrow +\infty} \left(\frac{ n+2 }{ n }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{n \rightarrow +\infty} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( n+2 \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( n \right) }\right)$

Find the derivative

$\lim_{n \rightarrow +\infty} \left(\frac{ 1 }{ \frac{ \mathrm{d} }{ \mathrm{d}n} \left( n \right) }\right)$

Find the derivative

$\lim_{n \rightarrow +\infty} \left(\frac{ 1 }{ 1 }\right)$

Any expression divided by $1$ remains the same

$\lim_{n \rightarrow +\infty} \left(1\right)$

The limit of a constant is equal to the constant

$1$