# Which values of x this function is continuous? (x non-negative real number and n->infinity)

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#f(x)=limlog(e^n+x^n)/n#

##### 2 Answers

For

and using the properties of logarithms:

Now, as

For

and again

For

We have now

We can conclude that the function

and is therefore continuous for every

It is continuous for all non-negative real

#### Explanation:

It looks like the function is:

In the following

For every

Apply l"Hospital's Rule (differentiate with respect to

**Case 1** For

#(1+(x/e)^nlnx)/(1+(x/e)^n# whose limit at#oo# is#1# .

**Case 2** For

**Case 3** For

#((e/x)^n+lnx)/((e/x)^n+1)# whose limit at#oo# is#lnx# .

So,