Math Problem Statement
lim(x -> 0+) sqrt^3(x) * lnx
Solution
To find the limit of the expression , we can break down the problem into parts.
We are evaluating:
Step 1: Analyze the behavior of each term as .
-
Cube Root of x:
. As , . -
Natural Logarithm of x:
tends to as .
Step 2: Combining the two terms.
We have an indeterminate form as . This suggests we need to analyze it more carefully, possibly using techniques such as L'Hopital's Rule, or rewriting the expression.
Step 3: Apply L'Hopital's Rule.
We can rewrite the expression as:
This is now in the form , which is an indeterminate form, and we can apply L'Hopital's Rule. To do that, we differentiate the numerator and denominator with respect to :
- Derivative of the numerator is .
- Derivative of the denominator is .
So, applying L'Hopital's Rule:
Step 4: Evaluate the limit.
As , . Therefore, the limit is:
Final Answer:
Would you like further explanation or have any questions? Here are some related questions:
- What happens to ?
- How does L'Hopital's Rule apply to other indeterminate forms like or ?
- How can I apply L'Hopital's Rule to limits involving exponential functions?
- Can you explain why as ?
- How would this limit change if the expression was ?
Tip: When you encounter limits of the form , it's often useful to rewrite them as fractions and apply L'Hopital's Rule.
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Math Problem Analysis
Mathematical Concepts
Limits
Indeterminate Forms
L'Hopital's Rule
Formulas
Cube root: sqrt^3(x) = x^(1/3)
Natural logarithm behavior: ln(x) → -∞ as x → 0+
L'Hopital's Rule: If lim(x → c) f(x)/g(x) = indeterminate form, then lim(x → c) f'(x)/g'(x)
Theorems
L'Hopital's Rule
Suitable Grade Level
Undergraduate Mathematics
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