00:01
Hello everyone, in this question we need to find the limit as n tends to infinity.
00:04
So, we have given four parts.
00:06
So, first part is we have given a n is equals to n cube plus n square plus 1 divided by 2 times of n cube minus 2 times of n plus 2.
00:21
Now, we need to take the limit.
00:24
So, limit n tends to infinity, this is equals to n cube plus n square plus 1 divided by 2 times of n cube minus 2 times of n plus 2.
00:35
Now, if we take limit, so we get infinity by infinity which is an indeterminate form.
00:40
So, we can rewrite the above equations that is limit n tends to infinity.
00:45
Now, we can take common n cube from the numerator and the denominator as well.
00:49
So, we have 1 plus 1 upon n plus 1 upon n cube divided by 2 minus 2 upon n square plus 2 upon n cube.
01:03
Now, if we take limit, we get 1 by 2 as our answer.
01:07
Now, this is the first part.
01:09
Now, for the second part we have given that is a n is equals to sin n upon n.
01:16
Now, we know that sin the range of sin x is minus 1 that is sin n comes under minus 1 to 1.
01:31
So, by squeeze theorem we can say that this is we can also write as this minus 1 upon n is less than equals to sin n upon n less than equals to 1 upon n.
01:45
Now, so if we take limit on all sides, so by squeeze theorem we can say that limit n tends to infinity sin n upon n is equals to 0.
01:59
Why this so? because if we take limit n tends to infinity of 1 by n, this comes out is equals to 0 and also if we take limit n tends to infinity of minus 1 by n, this is also equals to 0.
02:15
So, by squeeze theorem we can say that the limit n tends to infinity sin n upon n is also equals to 0.
02:22
So, this is the second part.
02:24
Now, for the third part we have given that is a n is equals to e to the power n upon 3 to the power n plus 2 to the power n.
02:34
So, now if we take limit n tends to infinity of the above expression that is e to the power n upon 3 to the power n plus 2 to the power n, we get that is e to the power infinity is infinity and 3 to the power infinity is also infinity...