sqrt(0+sqrt(0+sqrt(0+...))) = ?

TL;DR

Exploring the limits of sequences with square roots and continuous functions.

Transcript

okay as we all know square root of 0 is equal to 0 and the limit as x approaching 0 plus of square root of x that's also equal to 0. in fact this shows square root of x is continuous when x is equal to 0. but now have you ever thought about what if we have square root of zero plus square root of zero plus square root of zero plus dot dot and the li... Read More

Key Insights

• 🫚 Square root of zero is continuous at x=0, providing a basic understanding.
• ⛔ Limit of a sequence of continuous functions may result in a discontinuous function.
• 🫚 Recursive functions can help understand the limit of nested square roots.
• ⛔ The process of finding limits of sequences involves iterations and solving equations.
• ⛔ Understanding the domain of functions is crucial to interpreting limits correctly.
• 🔌 Limits can be evaluated by plugging in values and considering the domain restrictions.
• 😥 The fixed-point criteria test can be utilized to check the convergence of limits.

Questions & Answers

Q: Why is the square root of zero continuous at x=0?

The exact zeros in sequence lead to a continuous function at x=0.

Q: How does the limit of a sequence of continuous functions differ?

The limit may not be continuous due to x not being exactly equal to zero anymore.

Q: How are recursive functions used to understand the limit of nested square roots?

By defining a sequence of functions, like Yn = sqrt(x) + Yn-1, we can find the limit as n approaches infinity.

Q: Why is the limit for nested square roots equal to one?

By solving the limit equation iteratively, we find the quadratic equation leading to the limit being 1.

Summary & Key Takeaways

• Square root of zero is continuous at x=0.

• The limit of a sequence of continuous functions may not be continuous.

• Using recursive functions to understand the limit of nested square roots.