Formal Definition Of A Limit

Calculus Level 3

Find the limit that can be rewritten as $\\\forall \epsilon >0 \,\; \exists N>0:x>N\Rightarrow \left |f(x)-l \right |< \epsilon.$

\displaystyle{\lim_{x \to +\infty}f(x)=l}

\displaystyle{\lim_{x \to 0}f(x)=l}

\displaystyle{\lim_{x \to l}f(x)=0}

\displaystyle{\lim_{x \to -l}f(x)=0}

\displaystyle{\lim_{x \to l}f(x)=+\infty}

1 solution

Jacob Moore
Jul 17, 2018

Written out in English, this problems states that: "For all epsilon greater than zero, there exists a number $N$ ( $N$ is usually some arbitrarily large number) greater than zero so that $x$ being greater than $N$ ( $x$ is therefore approaching infinity since $N$ is already an arbitrarily large number) implies that $f(x)$ is approaching some limit $l$ and is less than epsilon away (for some value of $x$ greater than $N$ )" That being said, we can deduce for the statement given that as $x$ grows larger and larger, $f(x)$ will grow closer and closer to $l$ . Therefore implying $\lim_{x->infinity} (f(x))=l$

Formal Definition Of A Limit

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