1.Study the point convergence and the uniform convergence of the succession of functions with general term

We first study the pointwise limit for . It must be taken into account when calculate the limit that the variable x can take any value in the domain of the function. Recalling that the function is bounded in the interval , and also for is bounded in the open interval whenever we get:

Since when. In the following illustration we see the graphs of three terms: , in black, red and green, respectively, as the index increases the function tends to take smaller values:

Therefore is point-wise convergent to the null function . Let’s now look at the uniform convergence.

**Criterion of uniform convergence of a sequence**Converges uniformly to the point boundary whenever it is fulfilled: .

We apply it: . To find the maximum we derive and equate to zero, the absolute value does not concern us because the functions are symmetric respect to the abscissa axis (see previous figure):

The first case is discarded because it provides a minimum not a maximum, since . For the second case, using the equality and calling to the point at which we have a maximum:

We apply the limit:

Therefore the sequence converges uniformly.

2.Study the pointwise convergence and the uniform convergence of the sequence of functions with general term Also study the uniform convergence in the constrained interval .

For small values of the fraction takes large values, and the , whereas for large values the opposite happens, . In the following image we have the graph of (in the point the image is not shown, which is zero):

The transition point between the two values of can be easily found: ; At point we have , and we can then redefine the function definition: