Calculate limit (x^2+2x+1)/(2x^2-2x-1), x->+infinity
limit (x^2+2x+1)/(2x^2-2x-1) when x--> inf
We observe that he highest power for the numerator and...
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The limit `lim_(x->oo)(x^2+2x+1)/(2x^2-2x-1)` has to be determined.
Substituting `x = oo` in the given expression `(x^2+2x+1)/(2x^2-2x-1)` gives the result `oo/oo` which is indeterminate. If the result obtained while determining limits is of the form `oo/oo` or `0/0` it is possible to use l'Hospital's rule and substitute the numerator and denominator by their derivatives.
The derivative of (x^2+2x+1) is 2x + 2 and the derivative of (2x^2-2x-1) is 4x - 2.
The given limit can be written as `lim_(x->oo) (2x + 2)/(4x - 2)`
If we substitute `x = oo` , we again get the indeterminate form `oo/oo` . Continue the earlier step and again substitute the denominator and numerator with their derivative.
This gives 2/4 = 1/2
The required limit `lim_(x->oo)(x^2+2x+1)/(2x^2-2x-1) = 1/2`