x^sqrt(x) when x is 0 vs. x is approaching 0

TL;DR

X to the power of square root of X approaches 0 when X is exactly 0, but when approaching 0 from the right, the limit is 1.

Transcript

okay in this video that's focused on X to the square root of x power when X is exactly equal to 0 and when X is approaching 0 on the right hand side namely with hit the limit as X approach 0 + and let's do this one right here first when we have to calculate f of 0 this right here you have to remember the Romans will have exact 0 so you are going to... Read More

Key Insights

• ✊ Computing X to the power of square root of X at X=0 results in an undefined value for 0 to the power of 0.
• 👉 When calculating the limit as X approaches 0 from the right, X to the square root of X power simplifies to 1.
• ⛔ In a limit question where 0 to the 0 power is an indeterminate form, L'Hopital's Rule can be used with natural logarithms to find the limit.
• ☺️ The base of X to the square root of X power can be rewritten as E to the square root of X times Ln X for easier computation.
• 👉 The limit as X approaches 0 from the right for X to the square root of X power is 1.
• ✊ The content emphasizes the difference between computing 0 to the 0 power in a regular computation and in a limit question.
• ✊ By understanding the concepts of indeterminate forms and using mathematical techniques, such as L'Hopital's Rule, limits involving 0 to the 0 power can be solved.

Questions & Answers

Q: What happens when computing X to the power of square root of X at X=0?

When plugging in X=0, the base becomes 0 and the power becomes square root of 0, resulting in 0 to the power of 0, which is undefined.

Q: What is the limit as X approaches 0 from the right for X to the square root of X power?

The limit as X approaches 0 from the right for X to the square root of X power is 1.

Q: How can we solve the indeterminate form of 0 to the 0 power in a limit question?

To solve this, use L'Hopital's Rule by taking the natural logarithm of the function and differentiating both numerator and denominator.

Q: Can we rewrite X to the square root of X power in a different form to compute the limit?

Yes, we can rewrite X to the square root of X power as E to the square root of X times Ln X, which changes the expression and allows us to compute the limit.

Summary & Key Takeaways

• When calculating f(0), with 0 as the base and square root of 0 as the power, the result is 0. However, 0 to the power of 0 is undefined.

• When calculating the limit as X approaches 0 from the right, the result is 1.

• To solve the indeterminate form of 0 to the 0 power in a limit question, use L'Hopital's Rule and natural logarithms.