Calculus: Limit & Continuity Video Lecture | Question Bank for GATE Computer Science Engineering - Computer Science Engineering (CSE)

Ans. In calculus, a limit is a fundamental concept that describes the behavior of a function as the input approaches a certain value. It represents the value that the function approaches or "tends to" as the input gets arbitrarily close to the given value.

Ans. To determine the limit of a function, you can evaluate the function at points close to the given value and observe the pattern of the resulting outputs. If the function approaches a single value as the input gets closer to the given value, that value is the limit. Alternatively, you can use algebraic techniques or graphical analysis to find the limit.

Ans. Continuity is a property of a function that implies the absence of any "breaks" or "holes" in its graph. A function is continuous at a point if its limit exists at that point and is equal to the value of the function at that point. In other words, a function is continuous if it can be drawn without lifting the pen from the paper.

Ans. To determine if a function is continuous, you need to check three conditions: - The function must be defined at the point in question. - The limit of the function as the input approaches the point must exist. - The limit of the function must be equal to the value of the function at that point. If all these conditions are satisfied, the function is continuous at that point.

Ans. Limits and continuity have various applications in computer science engineering, including: - Optimization algorithms: Limits can help in finding the optimum value of a function by analyzing its behavior as the input approaches certain values. - Numerical methods: Continuity is essential in numerical methods used to approximate solutions to equations or functions. - Graphics and image processing: Continuity allows for smooth rendering of images and graphics in computer programs. - Signal processing: Limits and continuity play a role in analyzing and processing signals in fields such as audio and video processing. - Machine learning: Limits and continuity are used in various mathematical models and algorithms employed in machine learning and data analysis.