Find the limit of 1+(2/(x-3))^(3x) as x approaches infinity

 Exercise

$\lim_{x\to\infty}\left(1+\left(\frac{2}{x-3}\right)^{3x}\right)$

 

Evaluate the limit $\lim_{x\to\infty }\left(1+\left(\frac{2}{x-3}\right)^{3x}\right)$ by replacing all occurrences of $x$ by $\infty $

$1+\left(\frac{2}{\infty -3}\right)^{3\cdot \infty }$

 

Any expression multiplied by infinity tends to infinity, in other words: $\infty\cdot(\pm n)=\pm\infty$, if $n\neq0$

$1+\left(\frac{2}{\infty -3}\right)^{\infty }$

 

Infinity plus any algebraic expression is equal to infinity

$1+\left(\frac{2}{\infty }\right)^{\infty }$

 

Any expression divided by infinity is equal to zero

$1+0^{\infty }$

 

Simplifying

$1$

$1$

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