Chapter 7: Q. 72 (page 605)

Prove that if $\{a_{k}\} \to L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) \to f (L)$ .

Short Answer

Expert verified

It is proved that if $\{a_{k}\} \to L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) \to f (L)$ .

Step by step solution

Step 1. Given Information

The objective is to prove that if $\{a_{k}\} \to L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) \to f (L)$ .

Step 2. Prove

As $f$ is a continuous function so for a $ε > 0$ there exist $δ > 0$ such as $|f (x) - f (L)| < ε \forall x \in N (L, δ)$ .

Now, $\lim_{k \to \infty} a_{k} = L$ there exists a $m \in N$ such that

$|a_{n} - L| < δ \forall n \geq m$ .

So, for all $n \geq m$ , $a_{n} \in N (L, δ)$ .

So, $|f (a_{n}) - f (L)| < ε$ for all $n \geq m$ .

Hence it is proved that $f (a_{k}) \to f (L)$ .

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Prove that if $\{a_{k}\} \to L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) \to f (L)$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove

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Prove that if ak→Land fis a function that is continuous at L, then fak→fL.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove

One App. One Place for Learning.

Most popular questions from this chapter

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Prove that if $\{a_{k}\} \to L$ and $f$ is a function that is continuous at $L$ , then $f (a_{k}) \to f (L)$ .