How do you find #lim (x+x^(1/2)+x^(1/3))/(x^(2/3)+x^(1/4))# as #x->oo# using l'Hospital's Rule?
You do not really need l'Hospital's rule as:
so as:
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To find lim (x+x^(1/2)+x^(1/3))/(x^(2/3)+x^(1/4)) as x approaches infinity using l'Hopital's Rule, we proceed as follows:
- Take the derivative of the numerator and denominator separately.
- Evaluate the limit of the resulting expressions as x approaches infinity.
- If the limit remains indeterminate (e.g., 0/0 or ∞/∞), repeat the process until a determinate form is obtained.
Let's apply l'Hopital's Rule:
First, find the derivatives:
Numerator: f'(x) = 1 + (1/2)x^(-1/2) + (1/3)x^(-2/3)
Denominator: g'(x) = (2/3)x^(-1/3) + (1/4)x^(-3/4)
Now, take the limit of the derivatives as x approaches infinity:
lim (x+x^(1/2)+x^(1/3))/(x^(2/3)+x^(1/4)) as x approaches infinity = lim (1 + (1/2)x^(-1/2) + (1/3)x^(-2/3)) / ((2/3)x^(-1/3) + (1/4)x^(-3/4)) as x approaches infinity
= lim ((1/2)x^(-1/2) + (1/3)x^(-2/3)) / ((2/3)x^(-1/3) + (1/4)x^(-3/4)) as x approaches infinity
Now, apply l'Hopital's Rule again:
= lim ((-1/4)x^(-3/2) - (2/9)x^(-5/3)) / ((-2/9)x^(-4/3) - (3/16)x^(-7/4)) as x approaches infinity
Since the degree of the numerator and denominator are the same, we can evaluate the limit by dividing the leading coefficients:
= (-1/4) / (-2/9) as x approaches infinity = 9/8
Therefore, lim (x+x^(1/2)+x^(1/3))/(x^(2/3)+x^(1/4)) as x approaches infinity equals 9/8.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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- How do you apply the ratio test to determine if #Sigma 1/n^3# from #n=[1,oo)# is convergent to divergent?
- How do you find #lim (e^t-1)/t# as #t->0# using l'Hospital's Rule?

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