Question 1192696
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The definition of a limit is  lim(x-->a) {{{ f(x) }}} = L    if:

        |f(x)-L| < {{{epsilon}}}   for  0 < |x-a| < {{{ delta }}}


So here, {{{delta}}} = 1, centered at x=0, so we need to evaluate f(-1) and f(1): 
    f(-1) = {{{sqrt((-1)^2 + 4) = sqrt(5) }}}
    f(1) = {{{sqrt((1)^2 + 4) = sqrt(5) }}}

So the exact value of {{{epsilon }}} is {{{ f(x)-L }}} = {{{ highlight( sqrt(5) - 2 ) }}}  or approx. 0.2361

Here, the function is symmetrical about the limit point x=0.  If the function was not symmetrical, you'd evaluate f(x) on both sides of the limit point, and you'd take the larger value of {{{epsilon}}}.