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## Homework Statement

Show that if f(x) is a function of slow growth on the real line,

[tex] \lim}_{\substack \varepsilon \rightarrow0^+} \langle f(x)e^{- \varepsilon |x|}, \phi (x) \rangle = \langle f, \phi \rangle [/tex]

where [itex] \phi (x) [/itex] is a test function.

## Homework Equations

Definition of distribution:

[tex] \langle f , \phi \rangle = \int_{R_n} f(x) \phi(x) dx [/tex]

## The Attempt at a Solution

If I look at the function [itex] f_\varepsilon (x) = f(x)e^{-\varepsilon | x |} [/itex], I can say that the function is locally integrable, (actually [itex] L_1 (-\infty, \infty) [/itex]). Can't I just invoke the Lebesgue Dominated Convergence theorem here? That is, since [itex] f_\varepsilon [/itex] is locally integrable, and [itex] f_\varepsilon \rightarrow f [/itex] pointwise, that [itex] f_\varepsilon \rightarrow f [/itex] in a distributional sense.

Or am I missing something here. Are there other considerations to take into account?