How do you solve a limit when it approaches infinity and there is a square root in the denominator? say the highest power is x^2 in both the numerator and denominator.
as x ---> inf
and the top and bottom have the same highest power, then those terms will determine the limit
See my answer to a similar question by James at 11:06
I understand that problem. I get confused with the square root in the denominator. Aren't you supposed to do something with absolute values?
did you mean something like
lim (4x^2 -2√x)/(3x^2 + 5√2x)
of course it could only approach + inf , or else the root terms would be undefined.
Other than that, the same rule applies
the limit would be 4/3
Yeah like
limit as x approaches infinity( 3x^2+2x)/(sqrt(x^2-2x)
Would it equal 3 then?
No,
notice the highest powers are NOT the same, so my explanation is not valid.
as x --> inf
√(x^2 - 2x) ---> x
e.g. pick x = 1000000 and use your calculator to see
so the numerator ---> 3x^2
the denominator ----> x
so the limit is inf.
Ok I understand it now. Thank you.
To solve a limit when it approaches infinity and there is a square root in the denominator, you can use a technique called rationalization. Here's how you can proceed:
1. Start with the given expression of the limit, where the highest power is x^2 in both the numerator and denominator.
lim(x -> ∞) [sqrt(a*x^2 + b*x + c) / sqrt(p*x^2 + q*x + r)]
2. Multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a square root expression is obtained by changing the sign between the terms.
lim(x -> ∞) [sqrt(a*x^2 + b*x + c) / sqrt(p*x^2 + q*x + r)] * [(sqrt(p*x^2 + q*x + r)) / (sqrt(p*x^2 + q*x + r))]
This step helps in rationalizing the denominator.
3. Simplify the expression obtained after multiplying.
lim(x -> ∞) [sqrt((a*x^2 + b*x + c) * (p*x^2 + q*x + r)) / (p*x^2 + q*x + r)]
4. Distribute and simplify the square root in the numerator.
lim(x -> ∞) [sqrt((a*p*x^4 + (a*q+b*p)*x^3 + (a*r+b*q+c*p)*x^2 + (b*r+c*q)*x + c*r)) / (p*x^2 + q*x + r)]
5. Now, consider the highest power term in the numerator and denominator (which is x^4) and neglect the rest of the terms.
lim(x -> ∞) [sqrt((a*p*x^4) / (p*x^2))]
Simplifying further:
lim(x -> ∞) [sqrt((a * p * x^2) / (p))]
6. Cancel out the common terms and simplify.
lim(x -> ∞) [sqrt(a * x^2)]
lim(x -> ∞) [x * sqrt(a)]
7. Finally, take the limit as x approaches infinity.
lim(x -> ∞) [x * sqrt(a)] = ∞
Thus, the limit approaches infinity when there is a square root in the denominator, and both the numerator and denominator have x^2 as the highest power.