Question

X^(1/x) x>0

Explain the shape of the graph by computing the limit as
x → 0+ and as x → ∞.

Answer 1

as x approaches positive infinity what does 1 / x approach?

Answer 2

negative infinity

Answer 3

0 i mean

Answer 4

no

think about values like 1/100 , 1/1000, 1/1000000 , 1 / 1000000000 etc

what do these get close to?

Answer 5

0

Answer 6

right
now how about x^0 - what does that equal?

Answer 7

1

Answer 8

exactly, so as x--> infinity the graph approaches 1 and will look a horizontal line at f(x) = 1 In mathematical terms there will be an asynptote at f(x)= 1

Answer 9

as x approaches 0 from above what happens to 1/x ?

Answer 10

it gets bigger approaches infinity

Answer 11

yes - eg when x = o.001 for example 1/x = 1000
but as x is small then x^(1/x) will also be small

note when x = 1 f(x) = 1

Answer 12

it gets smaller

Answer 13

graph will look something like