Copied below is the output of a Maple session intended to convince people that interchange of limit and integral is not always possible. This was done to "condition" people to expecting the necessity of verifying additional conditions, such as uniform convergence, whose presence would guarantee that the interchange of limit and integral was valid, resulting in equal answers.
A sequence of functions kn(x) defined on [0,1] is defined below. The last lines of the Maple session show that the limit of the integrals of the kn's is 1, while the integral of the limit is 0.
> limit(x+(1/n),n=infinity);
x
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> limit(x+(1/n),x=infinity);
infinity
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> int(x^2,x);
3
1/3 x
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> int(x^2,x=2..3);
19/3
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> evalf(int(x^2,x=2..3));
6.333333333
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> Pi;
Pi
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> evalf(Pi);
3.141592654
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> f:=x->1/(1+x^2);
1
f := x -> ------
2
1 + x
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> f(2);
1/5
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> plot(f(x),x=-5..5);
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> g:=x->(1/Pi)*f(x);
f(x)
g := x -> ----
Pi
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> evalf(int(g(x),x=-5000..5000));
.9998726760
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> h:=(x,t)->t*g(t*x);
h := (x,t) -> t g(t x)
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> evalf(int(h(x,5000),x=-1..1));
.9998726760
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> k:=(x,n)->h(x-(1/n),n^2);
2
k := (x,n) -> h(x - 1/n, n )
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> evalf(int(k(x,1000),x=0..1));
.9996813715
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> limit(int(k(x,n),x=0..1),n=infinity);
1
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> limit(k(x,n),n=infinity);
0
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>