## Properties Of Limits

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# Properties of Limits

We now state some properties of limits without prools.

Assume that lim f (x) and lim g (x) exist. Let c be any real number.

x→a x→a

Then the following are true:

1. lim [f (x) ± g (x )] = lim f (x) ± lim g (x)

x→a x→a x→a

That is, the limit of sum or difference is the sum or difference of the limits.

2. lim [f (x) g (x)] = lim f (x) . lim g (x)

x→a x→ a x→ a

That is, the limit of a product is the product of limits

lim f (x)

3. lim f (x) = x a , provided lim g (x) ≠ 0.

x-a g (x) lim g (x) x→ a

x →a

That is, the limit of a quotient is the quotient of the limits, provided the denominator

does not have a limit of 0.

4. lim [cf (x)] = c. lim f (x)

x→ a x → a

That is, the limit of constant times a function is the constant times the limit.

5. If h (x) = c for all x, then lim h (x) = c.

x →a

That is, the limit of a constant function is a constant. This is usually abbreviated by writing lim c = c.

x→ a

6. lim for any positive integer n.

7. lim x

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Assume that lim f (x) and lim g (x) exist. Let c be any real number.

x→a x→a

Then the following are true:

1. lim [f (x) ± g (x )] = lim f (x) ± lim g (x)

x→a x→a x→a

That is, the limit of sum or difference is the sum or difference of the limits.

2. lim [f (x) g (x)] = lim f (x) . lim g (x)

x→a x→ a x→ a

That is, the limit of a product is the product of limits

lim f (x)

3. lim f (x) = x a , provided lim g (x) ≠ 0.

x-a g (x) lim g (x) x→ a

x →a

That is, the limit of a quotient is the quotient of the limits, provided the denominator

does not have a limit of 0.

4. lim [cf (x)] = c. lim f (x)

x→ a x → a

That is, the limit of constant times a function is the constant times the limit.

5. If h (x) = c for all x, then lim h (x) = c.

x →a

That is, the limit of a constant function is a constant. This is usually abbreviated by writing lim c = c.

x→ a

6. lim for any positive integer n.

7. lim x

^{n}= a^{n}for any positive integer n.For more help in

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