RandomGuy1 said:Homework Statement
f(x) = [1 - tan(x)]/[1 - √2 sin(x)] for x ≠ π/4
= k/2 for x = π/4
Find the value of k if the function is continuous at x = π/4
The Attempt at a Solution
This means that lim x → π/4 f(x) = k/2
I put x = (π/4 + h) and then evaluated the limit as h tended to zero. Doesn't work. Get sin (2h) in the denominator. Can I get a hint?
Continuity is a mathematical concept that describes the smoothness or unbroken nature of a function or a curve. It means that there are no gaps, breaks, or jumps in the graph of the function, and it can be drawn without lifting the pencil from the paper.
A function is continuous if it satisfies the three conditions of continuity: 1) the function is defined at the point in question, 2) the limit of the function at that point exists, and 3) the limit is equal to the function value at that point.
There are three types of discontinuity: removable, jump, and infinite. Removable discontinuity occurs when there is a hole in the graph of the function, jump discontinuity occurs when there is a jump or gap in the graph, and infinite discontinuity occurs when the function approaches positive or negative infinity at a certain point.
The Intermediate Value Theorem states that if a continuous function has values of opposite signs at two points, then it must have a root (a point where the function equals zero) between those two points. In other words, a continuous function must cross the x-axis at least once between two points where the y-values have opposite signs.
Continuity is a fundamental concept in calculus and is used to solve real-world problems involving rates of change and optimization. It is also used in physics and engineering to describe the smoothness of physical phenomena, such as motion and heat transfer. Additionally, continuity is important in computer science and data analysis to ensure the accuracy and reliability of algorithms and models.