**Introduction to Properties of Limits**

To increase our facility in manipulating limits, we have certain arithmetical and functional rules about limits. Any of these may be verified using the rigorous definition of limit that was provided at the beginning of the last section. We shall state the rules and get right to the examples.

If *f* and *g* are two functions, *c* is a real number, and lim _{x → c} *f* ( *x* ) and lim _{x → c} *g* ( *x* ) exist, then

*Theorem 1*

*Theorem 1*

(a) lim _{x → c} ( *f* ± *g* )( *x* ) = lim _{x → c} *f* ( *x* ) ± lim _{x → c} *g* ( *x* );

(b) lim _{x → c} ( *f* · *g* ) ( *x* ) = (lim _{x → c} *f* ( *x* )) · (lim _{x → c} *g* ( *x* ));

(c)

(d) lim _{x → c} ( *α* · *f* ( *x* )) = α · (lim _{x → c} *f* ( *x* )) *for any constant α* .

Some theoretical results, which will prove useful throughout our study of calculus, are these:

*Theorem 2*

*Theorem 2*

*Let a < c < b. A function f on the interval {x : a < x < b} cannot have two distinct limits at c* .

*Theorem 3*

*Theorem 3*

*does not exist.*

*Theorem 4* (The Pinching Theorem)

*Theorem 4*(The Pinching Theorem)

*Suppose that f, g, and h are functions whose domains each contain S = (a, c) ∪ (c, b). Assume further that*

*g* ( *x* ) ≤ *f* ( *x* ) ≤ *h* ( *x* )

*for all x * S. Refer to Fig. 2.4.

*If*

*and*

*then*

**Examples**

**Example 1**

Calculate lim _{x → 3} 4 *x* ^{3} − 7 *x* ^{2} + 5 *x* − 9.

**Solution 1**

We may apply Theorem 2.1(a) repeatedly to see that

We next observe that lim _{x → 3} *x* = 3. This assertion is self-evident, for when *x* is near to 3 then *x* is near to 3. Applying Theorem 2.1(d) and Theorem 2.1(b) repeatedly, we now see that

Of course lim _{x → 3} 9 = 9.

Putting all this information into equation (*) gives

**Example 2**

Use the Pinching Theorem to analyze the limit

**Solution 2**

We observe that

−∣ *x* ∣ ≡ *g* ( *x* ) ≤ *f* ( *x* ) = *x* sin *x* ≤ *h* ( *x* ≡ ∣ *x* ∣.

Thus we may apply the Pinching Theorem. Obviously

We conclude that lim _{x → 0} *f* ( *x* ) = 0.

**Example 3**

Analyze the limit

**Solution 3**

The denominator tends to 0 while the numerator does not. According to Theorem 2.3, the limit cannot exist.

**You Try It:** Use the Pinching Theorem to calculate lim _{x → 0} *x* ^{2} sin *x* .

**You Try It:** What can you say about

Find practice problems and solutions for these concepts at: Foundations of Calculus Practice Test.

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