`lim_(x -> oo) (5x^2 + 2)/(sqrt(x^2 + 3))` Find the limit.

Expert Answers

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You need to evaluate the limit, hence, you need to replace `oo` for x in equation:

`lim_(x->oo) (5 x^2 + 2)/(sqrt(x^2+3)) = (5 oo + 2)/(sqrt(oo+3)) = (oo)/(oo)`

Since the result is indeterminate, you need to factor out` x^2` at numerator and `x^2` at denominator:

`lim_(x->oo)(x^2(5 + 2/x^2))/(sqrt(x^2(1 + 3/x^2)))`

`lim_(x->oo)(x^2(5 + 2/x^2))/(|x|sqrt(1 + 3/x^2))`

Since `lim_(x->oo) 2/(x^(3/2)) = 0` and` lim_(x->oo) 3/x^2 = 0,` yields:

`lim_(x->oo) (x^(2-1))*(5/1)= 5*lim_(x->oo) x=5*oo = oo`

Hence, evaluating the given limit yields `lim_(x->oo) (5 x^2 + 2)/(sqrt(x^2+3)) = oo.`

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