Prove the limit statement -Select the correct statement for the definition of the limit: $\lim _ { x \rightarrow 0 } f ( x ) = L$ means that___________ A) if given any number $\varepsilon 0$, there exists a number $\delta 0$, such that for all $x$, $0 0$, there exists a number $\delta 0$, such that for all $x$, $0 0$, such that for all $x$, $0

Question

Prove the limit statement -Select the correct statement for the definition of the limit: $\lim _ { x \rightarrow 0 } f ( x ) = L$ means that___________ A) if given any number $\varepsilon > 0$, there exists a number $\delta > 0$, such that for all $x$, $0 < \left| x - x _ { 0 } \right| < \varepsilon$ implies $| f ( x ) - L | > \delta$. B) if given any number $\varepsilon > 0$, there exists a number $\delta > 0$, such that for all $x$, $0 < \left| x - x _ { 0 } \right| < \varepsilon$ implies $| f ( x ) - L | < \delta$. C) if given a number $\varepsilon > 0$, there exists a number $\delta > 0$, such that for all $x$, $0 < \left| x - x _ { 0 } \right| < \delta$ implies $| f ( x ) - L | > \varepsilon$. D) if given any number $\varepsilon > 0$, there exists a number $\delta > 0$, such that for all $x$, $0 < \left| x - x _ { 0 } \right| < \delta$ implies $| f ( x ) - L | < \varepsilon$.

Quizplus · Accepted Answer

The correct definition of the limit is that for any given number $\varepsilon > 0$, there exists a number $\delta > 0$ such that if $0 < \left| x - x _ { 0 } \right| < \delta$, then $| f ( x ) - L | < \varepsilon$. This means that as x approaches $x _ { 0 }$, the values of f(x) approach the limit L, and get arbitrarily close to L as x gets arbitrarily close to $x _ { 0 }$. Option D is the only statement that correctly captures this definition.

Prove the Limit Statement -Select the Correct Statement for the Definition

Prove the Limit Statement
-Select the Correct Statement for the Definition