00:01
Hello, let's look at the question.
00:03
We have been given to evaluate the limit of first term means, x approaches infinity e to the power 4 minus 9x by 1 plus 3x.
00:15
Let's see how we can do this.
00:17
We can rewrite this limit expression as e to the power limit x approaches infinity 4 minus 9x by 1 plus 3x.
00:31
As x approaches infinity, the terms involving x in the numerator and denominator denoting the expression, therefore we can ignore the constant terms.
00:39
Let's see limit x approaches infinity 4 minus 9x by 1 plus 3x.
00:46
Limit x approaches infinity minus 9x by 3x and limit x approaches infinity will have minus 3.
01:00
Thus the limit is minus 3 for this expression.
01:02
Let's see the second one.
01:06
We have been given limit x as minus 1 under root x square plus 8 minus 3 by x square plus 6.
01:18
So we can simplify the expression by rationalizing the denominator.
01:23
We get limit x approaches minus 1 under root x square plus 8 minus 3 by x square plus x under root x square plus 8 plus 3 by x square plus 8 plus 3.
01:49
We rationalize the denominator.
01:51
This simplifies to limit x approaches minus 1 x square plus 8 minus 9 by x square plus x under root x square plus 8 plus 3.
02:08
Now we can substitute x equals to minus in the given expression.
02:12
Then we have minus 1 square plus 8 minus 9 minus 1 square minus 1 multiplied by under root minus 1 square plus 8 plus 3.
02:32
Simplifying this, we get minus 2 by minus 2 under root 9 plus 3 which is minus 2 by 2 multiplied by 6 which will be equals to minus 1 by 6.
02:48
Therefore the limit of this expression is minus 1 by 6.
02:51
Now let's look at the third part.
03:00
We have been given limit for x approaches infinity 1 minus 7 by x to the power x.
03:12
As our x approaches infinity, the expression inside the parenthesis approaches 1 and the exponent becomes in determinate form of 1 to the power infinite.
03:23
To evaluate this limit, we can rewrite the natural logarithm limit x up to infinity and in 1 minus 7 by x x.
03:33
Using the property of law, we can bring down the exponent as a coefficient.
03:38
So we can write limit x approaches infinity x multiplied by ln 1 minus 7 by x.
03:48
Now we can substitute as x is equals to 1 by t...