Question

I am confused about a limit problem as x --> 0+ that includes "2^(1/x)", because obviously if 0 gets plugged in, you can't solve it. Anyway I am not sure how to get rid of this before plugging in 0

Answer 1

as x goes to zero from the right, 1/x goes to positive infinity and therefore so does
\[2^{\frac{1}{x}}\]

Answer 2

you are absolutely right that it cannot be solved it 0 is plugged in.

although, the limit at x approaches 0 considers x as a ridiculously small number.

the smaller the number you divide 1 by, the larger the answer. we can consider (1/x) to equal infinity since x is infinitely small.. this is 2^(infinity) = infinity

Answer 3

the limit doesnt care about what the value is AT zero; just what its approaching

Answer 4

The complete problem is to find the limit of (1/(1+(2^(1/x)))) as x-->0+.

Answer 5

looks geometrical

Answer 6

\[\frac{1}{1+r^n}\]

Answer 7

oh good point

Answer 8

similarly and opposite to (1/x) as x-->0+ = infinty, the larger the number you divide 1 by, the smaller the answer is. 1/infinity is such a small number that we consider it = 0.

the complete problem is 1/infinity. therefore x-->0+ = 0

Answer 9

the limit as*

Answer 10

Can you clarify what you mean when you say "similarly and opposite to (1/x) as x-->0+ = infinity"

Answer 11

well, 1 divided by a small number gives a large number, and 1 divided by a large number gives a small number.

by considering the small number as the limit of x approaching 0, the number is infinitely small.

it's the opposite for the limit 1/x as x-->infinity.

Answer 12

the answer in the second case being equal to an infinitely small number, 0

Answer 13

That makes sense. Thank you very much. In most cases when we are not looking at limits as x-->inf, we multiply everything my something in both numerator and denominator while not changing the equation to get it into a form where if you enter the target number, here 0, we will not get something undefined. That is why I was confused about whether I needed to take a similar approach of rephrasing the question to yield a better form, but if it is solvable without that measure, that is great

Answer 14

"we" being my calculus class

Answer 15

*by something

Answer 16

Thank you very much. I though about your comments more, and this makes perfect sense!

Answer 17

my pleasure :) i appreciate the positive feedback

Answer 18

and i appreciate your detailed answers! :-)