Learn how to prove that a function is bounded with this step-by-step guide. Master the techniques for a bounded function proof.
Featured Articles on: LIMITS
Understanding Derivatives Simply: A Beginner’s Guide
Understanding derivatives simply involves grasping how functions change. This guide offers an intuitive explanation for beginners.
Understanding the Ramp Function: Definition and Applications
Explore the ramp function, its definition, properties, and applications in signal processing and control systems. Understand the ramp function today!
Arc Length Functions: A Deep Dive
Explore arc length functions and their cardinality. Discover how the number of continuous functions changes with increasing arc length.
Defining Integrals: Key Conditions Explained
Explore the conditions for defining integrals, focusing on the necessary structures of spaces and functions for meaningful integration.
Area of Recursive Functions: A Detailed Analysis
Explore the area of recursive functions with a detailed analysis. Discover how the area remains constant as the recursion depth increases.
Understanding Function Decay Rates: Slow vs. Rapid
Explore function decay rates and how the parameter α affects whether a function has slow or rapid decay. Learn with examples!
Constructing Continuous Functions: Examples and Proofs
Learn how to build functions continuous at specific points, like integers or irrationals. Explore examples and proofs for constructing continuous functions.
Understanding ##\Theta## Complexity: Proving Floor Function’s Growth
Explore proving that the floor function has Theta complexity of Θ(x). Learn about upper and lower bounds in this analysis.
Fractional Differentiability of Functions
Explore fractional differentiability functions and how they behave with non-integer derivatives. Learn about constructing functions with specific differentiability.
Finding Minimum Value Function
Learn how to find the minimum value function using calculus. Step-by-step guide included!
Density of Smooth Functions in L1 and L2 Spaces
Explore the density of smooth functions in L1 and L2 spaces. Learn how smooth functions approximate complex functions effectively.
Find the limit: \( \lim_{x \to 3} (2x + 5) \)
Find \( \lim_{x \to 3} (2x + 5) \) Solution:To solve this limit, we substitute the value of \( x \) directly because the function is continuous at \( x = 3 \).\( \lim_{x \to 3} (2x + 5) = 2(3) + 5 = 6 + 5 = 11 \)
Morning Refresher – 5 Basic Problems in Limits to Boost Your Mind
5 Basic Problems on Limits just to refresh your mind. Problem 1 Find the limit: \( \lim_{x \to 2} (3x – 4) \) Solution: To solve this limit, we substitute the value of \(x\) directly because the function is continuous at \(x = 2\). \[ \lim_{x \to 2} (3x –…
THEOREM# \( \lim_{\theta\to0} \dfrac{sinθ}{θ} \) = 1
We have \( \lim_{\theta\to0} { \sin\theta \over \theta } \) = 1 Consider the below diagram. We have r = radius of the circle.A = centre of the circle.The sector ⌔ formed by the arc BD subtends an angle θ at the centre. Case 1 : θ > 0 i.e. θ is +ve Let 0 ≤ θ ≤ \(…
Theorem# \( \lim_{x \to a} { x^n – a^n \over x – a } = na^{n-1} \)
To prove : lim\( _{x \to a} { x^n – a^n \over x – a } = na^{n-1} \) where n is a rational number Proof: Let \( x = a + h \) Then as \(x \to a \), we have \(h \to 0 \) Now, \( \lim_{x \to a} { x^n – a^n \over x – a }…