How to test continuty of this function ? f(x) = (1+x)^(1+x)(1+x)1+x x is not equal zero

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2 Answers
Andrea S.
Sep 29, 2017

f(x)f(x) is continuous for every x in (-1,+oo)x∈(−1,+∞).

Explanation:

Write the function as:

f(x) = e^((1+x)ln(1+x))f(x)=e(1+x)ln(1+x)

As the exponential function e^tet is continuous for any t in RR, and (1+x)ln(1+x) is continuous for x > -1 then f(x) is continuous for every x in (-1,+oo) and x!=0

For x=0:

lim_(x->0) f(x) = lim_(x->0) e^((1+x)ln(1+x)) = e^((lim_(x->0)(1+x)ln(1+x))) = e^0 =1 = f(0)

So the function is continuous also for x = 0

graph{(1+x)^(1+x) [-10, 10, -5, 5]}

yumean1119
Sep 29, 2017

This function is not continuous at x=0

Explanation:

Consider a function g(h)=(1+1/h)^h.

This function approrches e(=2.718…) when h-> oo (by definition of e). This is also true when h goes to negative infinity.

So, the limit of f(x) when x approaches 0 is
lim_(x->+0) f(x) = lim_(h->oo) g(h) =e
lim_(x->-0) f(x) = lim_(h->-oo) g(h) =e

They are different from f(0) and therefore f(x) has a jump
discontinuity at x=0.

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