voodoochild said:Homework Statement
Let r (t)=f(t),g(t),h(t)and s(t)=〈F(t),G(t),H(t)〉.
Show that lim(t→a)(r (t)+s (t))=lim(t→a)[r (t)]+lim(t→a)[s (t)].
Homework Equations
The Attempt at a Solution
I know that if a function r = <f,g,h> and lim(t→a)[r(t)] then lim(t→a)[r(t)] = < lim(t→a)[f(t)], lim(t→a)[g(t)], lim(t→a)[h(t)] >
A vector-valued function is a mathematical function that takes in one or more inputs and returns a vector as its output. Unlike scalar functions, which return a single numerical value, vector-valued functions return a set of numbers arranged in a specific order.
Some common examples of vector-valued functions include parametric equations, where the x, y, and z coordinates of a point are described in terms of a parameter, and velocity and acceleration functions, which describe the rate of change of position and velocity, respectively, over time.
Vector-valued functions are typically represented using a single variable, such as t, and a set of equations that describe the relationship between that variable and the vector components. For example, a 2D vector-valued function could be represented as (x(t), y(t)), where x and y are functions of t.
The domain of a vector-valued function is the set of all possible input values, while the range is the set of all possible output vectors. In other words, the domain is the set of values for the independent variable, and the range is the set of resulting vectors for those inputs.
Vector-valued functions are used in various fields of science and engineering, such as physics, computer graphics, and economics. They are particularly useful for describing complex systems that involve multiple variables and can help predict the behavior of these systems over time.