Pointwise Limit & Uniform Convergence of fn(x)=ne^(-xn^2)

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SUMMARY

The sequence of functions defined as fn(x)=ne^(-xn^2) converges pointwise to the limit function f(x) = 0 for all x > 0, while f(0) is undefined due to divergence at that point. The convergence is not uniform because the limit function f(x) lacks continuity, contrasting with the continuity of each function in the sequence fn(x). This distinction confirms that uniform convergence does not occur in this case.

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For the sequence of functions fn(x)=ne^(-xn^2) on [0,infinity), what is the pointwise limit of this sequence?is the converbence uniform?
 
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Limit functon f(x) is:

f(x) = 0

for x > 0

f(0) is not defined because for x = 0 the sequence diverges.

Because f(x) is not continuous while all the f_n(x) are continuous, the convergence is not uniform.
 

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