Chapter 1: Q. 89 (page 151)

Prove that the sum rule for limits also applies for limits as $x \to \infty$ : If $\lim_{x \to \infty} f (x) = L$ and $\lim_{x \to \infty} g (x) = M$ , then $\lim_{x \to \infty} (f (x) + g (x)) = L + M .$

Short Answer

Expert verified

It is proved that If $\lim_{x \to \infty} f (x) = L$ and $\lim_{x \to \infty} g (x) = M$ , then $\lim_{x \to \infty} (f (x) + g (x)) = L + M .$

Step by step solution

Step 1. Given Information

We are given two functions $f (x) and g (x)$ .

Step 2. Proving the statement

Given $ε > 0$ , we can choose $N_{1} > 0$ to get $f (x)$ within $\frac{ε}{2}$ of L and also choose $N_{2} > 0$ to get $g (x)$ within $\frac{ε}{2}$ of M,

Then for $N = \max (N_{1}, N_{2})$ and role="math" $x > N$ we have,

$(L + M) - ε < f (x) + g (x) < (L + M) + ε$

Hence, the sum rule for limits also applies for limits as $x \to \infty$ . If $\lim_{x \to \infty} f (x) = L$ and role="math" $\lim_{x \to \infty} g (x) = M$ , then role="math" localid="1648286593808" $\lim_{x \to \infty} (f (x) + g (x)) = L + M$ .

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Prove that the sum rule for limits also applies for limits as $x \to \infty$ : If $\lim_{x \to \infty} f (x) = L$ and $\lim_{x \to \infty} g (x) = M$ , then $\lim_{x \to \infty} (f (x) + g (x)) = L + M .$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

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Prove that the sum rule for limits also applies for limits as x→∞: If limx→∞f(x)=Land limx→∞g(x)=M, thenlimx→∞(f(x)+g(x))=L+M.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Geometry

Mechanics Maths

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Prove that the sum rule for limits also applies for limits as $x \to \infty$ : If $\lim_{x \to \infty} f (x) = L$ and $\lim_{x \to \infty} g (x) = M$ , then $\lim_{x \to \infty} (f (x) + g (x)) = L + M .$