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Question

Solve $$\displaystyle \lim_{n\rightarrow \infty} \dfrac{(n+1)^{2}+5n}{n^{2}}$$


A
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B
0
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C
1
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D
7
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Solution

The correct option is C $$1$$
We need to find value of limit $$ \lim _{ n\to \infty  } \dfrac { \left( n+1 \right) ^{ 2 }+5n }{ n^{ 2 } } $$
Apply Fraction Rule $$\dfrac { a+b }{ c } =\dfrac { a }{ c } +\dfrac { b }{ c }$$
$$=\lim _{ n\rightarrow \infty  }{ \left[ \dfrac { \left( n+1 \right) ^{ 2 } }{ { n }^{ 2 } } + \dfrac { 5n }{ { n }^{ 2 } }  \right]  }$$
$$=\lim _{ n\rightarrow \infty  }{ \left[ { \left( 1+\dfrac { 1 }{ n }  \right)  }^{ 2 }+ \dfrac { 5n }{ { n } }  \right]  }$$
Put the value $$n=\infty$$
$$=\left( 1+\dfrac { 1 }{ \infty  }  \right) ^{ 2 }+\dfrac { 5 }{ \infty  }$$
$$=1$$     ....As we know $$\dfrac {1}{\infty }=0$$

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