Question

what is limit (t approaches 0) of 3^(2/t) ? Please help, im stuck

Answer 1

\(\bf lim_{t \to 0} 3^{\frac{2}{t}}\\
\textit{if t becomes closer to 0 from the right } \quad t \to 0^+\\
t = \cfrac{1}{10}, \cfrac{1}{100},\cfrac{1}{1000},\cfrac{1}{10000},...\\
3^{\frac{2}{t}} \implies 3^{2\cdot 10}, 3^{2\cdot 100}, 3^{2\cdot 1000}, 3^{2\cdot 1000},...\\\quad \\\quad \\
\textit{if t becomes closer to 0 from the right } \quad t \to 0^-\\
t = -\cfrac{1}{10}, -\cfrac{1}{100},-\cfrac{1}{1000},-\cfrac{1}{10000},...\\
3^{\frac{2}{t}} \implies 3^{2\cdot -10}, 3^{2\cdot -100}, 3^{2\cdot -1000}, 3^{2\cdot -1000},...\\
3^{\frac{2}{t}} \implies \cfrac{1}{3^{20}}, \cfrac{1}{3^{200}}, \cfrac{1}{3^{2000}}, \cfrac{1}{3^{20000}}, ...\)

Answer 2

hmm, I meant from the left, anyhow :) \(\bf \textit{if t becomes closer to 0 from the left } \quad t \to 0^-\\
t = -\cfrac{1}{10}, -\cfrac{1}{100},-\cfrac{1}{1000},-\cfrac{1}{10000},...\\
3^{\frac{2}{t}} \implies 3^{2\cdot -10}, 3^{2\cdot -100}, 3^{2\cdot -1000}, 3^{2\cdot -1000},...\\
3^{\frac{2}{t}} \implies \cfrac{1}{3^{20}}, \cfrac{1}{3^{200}}, \cfrac{1}{3^{2000}}, \cfrac{1}{3^{20000}}, ...\)

Answer 3

You have to look at this separately from each side of 0.

If t is positive and t->0, then 2/t -> infinty, so what happens to 3^(2/t)?

If t is negative and t->0, then 2/t is negative, so 3^(2/t) = (1/3)^(2/|t|).
Then since t->0, (2/|t|)-> infinity, so what happens to the value of 3^(2/t)?

Answer 4

as DebbieG said, check the one-sided limit
notice as "t" goes to 0, where it's going

Answer 5

hmm, a couple of typos... anyhow \(\bf lim_{t \to 0} 3^{\frac{2}{t}}\\
\textit{if t becomes closer to 0 from the right } \quad t \to 0^+\\
t = \cfrac{1}{10}, \cfrac{1}{100},\cfrac{1}{1000},\cfrac{1}{10000},...\\
3^{\frac{2}{t}} \implies 3^{2\cdot 10}, 3^{2\cdot 100}, 3^{2\cdot 1000}, 3^{2\cdot 10000},...\\\quad \\\quad \\
\textit{if t becomes closer to 0 from the left } \quad t \to 0^-\\
t = -\cfrac{1}{10}, -\cfrac{1}{100},-\cfrac{1}{1000},-\cfrac{1}{10000},...\\
3^{\frac{2}{t}} \implies 3^{2\cdot -10}, 3^{2\cdot -100}, 3^{2\cdot -1000}, 3^{2\cdot -10000},...\\
3^{\frac{2}{t}} \implies \cfrac{1}{3^{20}}, \cfrac{1}{3^{200}}, \cfrac{1}{3^{2000}}, \cfrac{1}{3^{20000}}, ...\)

Answer 6

\[\lim_{t \rightarrow0 }( 5^t +3^\left\{ 2/t \right\} +2)\]

Answer 7

i was stuck there and i dont think seperating the limits is going to work. But i dont know how to solve it

Answer 8

well, the same is true for the whole expression
as "t" comes closer from the RIGHT, \(\bf 5^t\) becomes big and ADDS to \(\bf 3^{\frac{2}{t}}\) going to \(\bf +\infty\)

and the same will happen to \(\bf 5^t when "t" comes closer from the LEFT

Answer 9

the answer on the back of the book is 3. Though i still dont understand how they got it :)

Answer 10

hmmmm, well, I just gave it a graph, and that's is TRUE for the one-sided from the left, ONLY

Answer 11

Are you sure the problem is not
\[ \lim_{t \rightarrow 0^- }( 5^t +3^\left\{ 2/t \right\} +2) \]

are you leaving off the - from 0- ?