Applications of uniform convergence

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Consider the sequences #f_n# #(n=1,2,\ldots)# of real functions on the whole line given by one of the function rules below. Their pointwise limits exist but the convergence is not uniform. Drag the functions to the column below the reason for this fact.

#f_n(x)={{x+{{1}\over{n}}}\over{\left| x\right| +{{1}\over{n}}}}#

#f_n(x) = {{n\cdot x}\over{x^2+n}}#

#f_n(x) ={{x^2+n\cdot x+1}\over{x^2+x+n}}#

#f_n(x)={{\euler^{n\cdot x}-\euler^ {- n\cdot x }}\over{\euler^{n\cdot x}+\euler^ {- n\cdot x }}}#

#f_n(x)=\arctan \left(n\cdot x\right)#

#f_n(x)={{n\cdot \euler^{x}}\over{n\cdot \euler^{2\cdot x}+1}}#

Sequences whose terms are continuous but whose limit is not

Sequences whose terms are bounded but whose limit is not

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