Answer to Question #310402 in Real Analysis for Nikhil

Question

For all even integral value of n, lim (x+1)^-n

n to ∞

Exist or not

True or false with full explanation

Answer 1

$\mathop {\lim }\limits_{n \to \infty } {\left( {x + 1} \right)^{ - n}}$

$= \mathop {\lim }\limits_{n \to \infty } \exp \left( {\ln \left( {{{\left( {x + 1} \right)}^{ - n}}} \right)} \right)$

$= \mathop {\lim }\limits_{n \to \infty } \exp \left( { - n\ln \left( {x + 1} \right)} \right)$

$= \exp \left( { - \ln \left( {x + 1} \right) \cdot \mathop {\lim }\limits_{n \to \infty } n} \right)$

Now for $\ln \left( {x + 1} \right) > 0$

$\mathop {\lim }\limits_{n \to \infty } {\left( {x + 1} \right)^{ - n}}$

$= \exp \left( { - \ln \left( {x + 1} \right) \cdot \infty } \right)$

$= \exp \left( { - \infty } \right)$

$=0$

Hence $\mathop {\lim }\limits_{n \to \infty } {\left( {x + 1} \right)^{ - n}}=0$ for $\ln \left( {x + 1} \right) > 0$