We should consider the following properties of limits.
>>` lim_(x-gtoo) = cx^n = oo`
>> `lim_(x-gtgtoo) 1/x^n = 0`
>> If `lim_(x-gtc) f(x) ``= oo ` and ` lim_(x-gtc) = L` (where L<0),
then ` lim_(x-gtc) [f(x) g(x)]= -oo` .
To evaluate the limit above, factor out the x with largest power.
`lim_(x-gtoo) x^5(1/x^5 +2/x^4 - 3)`
Then, take the limit of each factor.
`lim_(x-gtoo) x^5 = oo` and `lim_(x-gtoo) (1/x^5+2/x^4-3) = 0+0-3 = -3`
Hence, the value of the limit is:
`lim_(x-gtoo) (1+x^2-3x^5) = -oo`
The limit `lim_(x->oo) 1 + 2x - 3x^5` has to be determined.
Notice the three terms in the expression 1 + 2x - 3x^5.
As x tends to `oo` , 1 does not change, 2x increases and 3x^5 also increases. But the rate at which 3x^5 increases is faster than that at which 2x increases as x has been multiplied by a larger number and also because x has been raised to a higher power.
As a result -3x^5, approaches `-oo` at a faster rate than at which 2x approaches `oo` .
This gives the limit `lim_(x->oo) 1 + 2x - 3x^5 = -oo` .