The sequence of functions converges to the function
.
Step-by-step explanation:
The limit exists and converges to zero whenever . But, if the sequence is constant and all its terms are equal to , then converges to . Using this result, consider the sequence of functions defined on the interval by . Then, for all we have that . Now, if , then . Therefore, the limit function of the sequence of functions is
.
To show that the convergence is not uniform consider . For any choose such that . Then
Check the picture below, so the parabola looks more or less like so, with a vertex at (0 , -7), let's recall the vertex is half-way between the focus point and the directrix.
so this horizontal parabola opens up to the left-hand-side, meaning that the "P" distance is a negative value.