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NWeid1
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Find the limit of x^2(√(x^4+5)-x^2) as x->∞. I think it might be a L'Hospitals rule, but I'm not sure. We haven't done many problems like this yet, thanks!
NWeid1 said:Find the limit of x^2(√(x^4+5)-x^2) as x->∞. I think it might be a L'Hospitals rule, but I'm not sure. We haven't done many problems like this yet, thanks!
Right. L'Hopital's Rule is not applicable here.SteamKing said:L'Hospital's Rule is valid only for certain indeterminate forms, like infinity/infinity or 0/0.
I don't know about that. This limit is another of the indeterminate forms - [∞ * 0].SteamKing said:You should be able to determine this limit by inspection.
The limit of x^2(√(x^4+5)-x^2) as x approaches infinity is infinity. This means that as x gets larger and larger, the value of the expression also gets larger and larger without bound.
To solve for the limit, we can use algebraic manipulation and the properties of limits. First, we can simplify the expression by multiplying the numerator and denominator by √(x^4+5)+x^2, which will cancel out the square root. Then, we can use the limit properties to split the expression into two separate limits. The limit of x^2 as x approaches infinity is infinity, and the limit of √(x^4+5)+x^2 as x approaches infinity is also infinity. Therefore, the overall limit is the product of these two infinities, which is infinity.
Yes, the limit of x^2(√(x^4+5)-x^2) as x approaches infinity is equal to the limit of (√(x^4+5)-x^2) as x approaches infinity. This is because when x is approaching infinity, the x^2 term becomes insignificant compared to x^4, so it can be ignored in the limit calculation.
No, L'Hospital's rule cannot be used to find the limit of this expression. L'Hospital's rule is used for finding limits of indeterminate forms, such as 0/0 or ∞/∞, but the expression x^2(√(x^4+5)-x^2) does not fall into any of these forms.
Finding the limit of this expression as x approaches infinity helps us understand the behavior of the function as x gets larger and larger. In this case, we can see that the function is growing without bound and has no finite limit. This can be useful in various applications, such as in predicting the behavior of a system as a variable approaches infinity.