1. lim _(n arrow infty)(sqrt(n^2)+5 n+1-sqrt(n^2)+3 n-1)

Question

Answer 1

To find the limit of the given expression as

approaches infinity, we can use the technique of rationalizing the numerator. Here's the step-by-step solution:Given expression:

First, we multiply and divide the expression by the conjugate of the numerator:

The numerator simplifies as follows:

So the expression becomes:

Next, we factor out

from the square roots in the denominator:

As

approaches infinity, the terms

and

in the first square root, and

and

in the second square root, approach zero. Therefore, the denominator simplifies to:

So the expression simplifies to:

Therefore, the limit is:

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