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11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275361

Chapter 1.5, Problem 70E

Textbook Problem

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Proof Prove the difference, product, and quotient properties in Theorem 1.15.

To determine

**To prove:** The properties of infinite limits.

**Difference:**

**Product:**

**Quotient:**

**Given:** The provided functions are

**Proof:**

Let *c* (except possibly at

Apply the definition of infinite limits:

For each

And since

So, there exists

Now, first show that the limit of

Let

Rewrite it as:

So,

Now, subtract both the inequality, you can write

Therefore,

Hence,

Now, show that the limit of product of functions:

**Case I:**

Choose M > 0. So, find

Let

So, there exists

By letting

Rewrite it as:

So,

Implies,

Now, product of both the inequality gives:

Therefore,

Hence,

**Case II:**

Choose M > 0. So, find

Let

So, there exists

By letting

Rewrite it as:

So,

Since

Take minus of the above inequality:

Now, product of both the inequality gives

Calculus of a Single Variable

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