Theorem 4.1.11: Sequential Criterion for Limits Let $$ f: A ightarrow \mathbb{R} $$ be a function and let $$ c $$ be a cluster point of $$ A $$. Then $$ \left(\lim _{x ightarrow c} f(x)=L ight. $$ if and only if for every sequence $$ \left(x_{n} ight) $$ in $$ A $$ converging to $$ c $$ (with $$ x_{n}
eq c $$ for all $$ \left.n \in \mathbb{N} ight) $$ the sequence $$ \left(f\left(x_{n} ight) ight) $$ converges to $$ L $$.

Question

Theorem 4.1.11: Sequential Criterion for Limits Let $$ f: A ightarrow \mathbb{R} $$ be a function and let $$ c $$ be a cluster point of $$ A $$. Then $$ \left(\lim _{x ightarrow c} f(x)=Light. $$ if and only if for every sequence $$ \left(x_{n}ight) $$ in $$ A $$ converging to $$ c $$ (with $$ x_{n} 
eq c $$ for all $$ \left.n \in \mathbb{N}ight) $$ the sequence $$ \left(f\left(x_{n}ight)ight) $$ converges to $$ L $$.

Alexander Page · Accepted Answer

**المعيار المتتالي للحدود** إذا كانت قيمة الدالة $$ f(x) $$ عند النقطة $$ c $$ تقترب إلى $$ L $$ عندما يقترب $$ x $$ من $$ c $$، فإن ذلك يعني أن جميع المتتاليات $$ x_n $$ التي تقترب إلى $$ c $$ (مع $$ x_n 
eq c $$) ستؤدي إلى قيمة $$ f(x_n) $$ تقترب إلى $$ L $$.

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