Question

Use the limit theorems for sequences to evaluate limn→[infinity]n+1−−−−−√−n−−√=0

Answer 1

To evaluate the limit lim(n→∞) (√(n+1) - √n) = 0, we can use the limit theorems for sequences. Firstly, we can rationalize the numerator by multiplying the expression by (√(n+1) + √n), giving us [(√(n+1) - √n)(√(n+1) + √n)]. This simplifies to [(n+1) - n], which is equal to 1.

Step-by-step explanation:

To evaluate the limit lim(n→∞) (√(n+1) - √n) = 0, we can use the limit theorems for sequences. Firstly, we can rationalize the numerator by multiplying the expression by (√(n+1) + √n), giving us [(√(n+1) - √n)(√(n+1) + √n)]. This simplifies to [(n+1) - n], which is equal to 1. Next, we can use the limit of a constant multiplied by a sequence, which states that if the limit of the sequence is L, then the limit of k times the sequence is kL. In this case, k is 1, so the limit of 1 times the sequence is 1 times 0, which equals 0.