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My instructor gave us a challenge problem to solve in limits: ##\lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 }\left( 1-\cos { \frac { 5.1 }{ x } } \right) \right] } ## Note that we did not take Hospital's rule yet so we couldn't have used it.

Now my first thought was to use the product law of limits to get:

##\lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 }\left( 1-\cos { \frac { 5.1 }{ x } } \right) \right] } =\lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 } \right] } \cdot \lim _{ x\rightarrow \infty }{ \left[ \left( 1-\cos { \frac { 5.1 }{ x } } \right) \right] } \\ \lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 }\left( 1-\cos { \frac { 5.1 }{ x } } \right) \right] } =\lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 } \right] } \cdot 0\\ \lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 }\left( 1-\cos { \frac { 5.1 }{ x } } \right) \right] } =0##

But when I pressed submit, it gave me an incorrect answer. I tried playing with it algebraically and I ended up giving up and having to use L'Hospital rule, even though we haven't taken it yet. I did:

##\lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 }\left( 1-\cos { \frac { 5.1 }{ x } } \right) \right] } =\lim _{ x\rightarrow \infty }{ \left[ \frac { 1-\cos { \frac { 5.1 }{ x } } }{ \frac { 1 }{ { x }^{ 2 } } } \right] } \\ Using\quad Hospital's\quad rule:\\ \lim _{ x\rightarrow \infty }{ \left[ \frac { 1-\cos { \frac { 5.1 }{ x } } }{ \frac { 1 }{ { x }^{ 2 } } } \right] } =\lim _{ x\rightarrow \infty }{ \left[ 2.55x\sin { \frac { 5.1 }{ x } } \right] } \\ using\quad Hospital's\quad rule\quad again:\\ \lim _{ x\rightarrow \infty }{ \left[ 2.55x\sin { \frac { 5.1 }{ x } } \right] } =\lim _{ x\rightarrow \infty }{ \left[ 2.55\cdot 5.1\cdot \cos { \frac { 5.1 }{ x } } \right] } \\ =\lim _{ x\rightarrow \infty }{ \left[ \frac { 2601 }{ 200 } \sin { \frac { 5.1 }{ x } } \right] } =\frac { 2601 }{ 200 } ##

Can anyone tell me why the first method didn't give me the correct answer? Also any hints about how to solve this algebraically or using any way other than L'Hospotal's rule?

Thanks

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# Using different limit laws give different answer?

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