Using l'Hospital's rule, determine the limit:`lim_(x-gtoo) ` `10x^4 - 2x^3` ?
The limit `lim_(x->oo) 10x^4 - 2x^3` has to be determined.
If the substitution `x = oo` is made, the result is `10*oo - 2*oo = oo - oo` . This is not equal to 0, rather it is indeterminate. l'Hopital's rule can be used in the indeterminate forms `0/0` and `oo/oo` . It is possible to convert the given limit to that form.
`lim_(x->oo) 10x^4 - 2x^3`
= `lim_(x->oo) (1/(2x^3) - 1/(10x^4))/(1/(10x^4*2x^3))`
But that does help in any way as it can be seen that irrespective of how many times the derivative of the numerator and the denominator is taken, the indeterminate form `0/0` remains.
To determine `lim_(x->oo) 10x^4 - 2x^3` , write it as:
`lim_(x->oo) x^3(10x - 2)`
If `x = oo` , `10x - 2 = oo`
The substitution `x = oo` gives the result `oo*oo` . This is equal to `oo` .
The required limit `lim_(x->oo) 10x^4 - 2x^3 = oo`