Question

Fill in the blanks to complete each of the following theorem statements.

Basic Limit Rules for Convergent Sequences: If $\left\{a_k\right\}\mathrm{and}\left\{b_k\right\}\mathrm{are}\mathrm{convergent}\mathrm{sequences}\mathrm{with}{a}_k\to L\mathrm{and}{b}_k\to M\mathrm{as}k\to \infty$and if c is any constant, then

If f is a function that is _____ at L, then$f\left(a_k\right)\to$

Short answer

The required answer is f is a function that is continuous at L, then$f\left(a_k\right)\to f\left(L\right)$

Step-by-step solution

Step 1. Given Information  

The given data is if$\left\{a_k\right\}\mathrm{and}\left\{b_k\right\}\mathrm{are}\mathrm{convergent}\mathrm{sequences}\mathrm{with}{a}_k\to L\mathrm{and}{b}_k\to M\mathrm{as}k\to \infty$

Step 2. Explanation   

Using the concept of limits of functions of sequences, that is,

If $a_k\to L$and f is a function that is continuous at L then$f\left(a_k\right)\to f\left(L\right)$