Properties of Limits

Notation of Limit

The limit of a function is designated by f (x) → L as x → a or using the limit notation:

lim_{x \to a} f (x) = L .

Below we assume that the limits of functions $lim_{x \to a} f (x),$ $lim_{x \to a} g (x),$ $lim_{x \to a} f_{1} (x),$ $\dots,$ $lim_{x \to a} f_{n} (x)$ exist.

Sum Rule

This rule states that the limit of the sum of two functions is equal to the sum of their limits:

lim_{x \to a} [f (x) + g (x)] = lim_{x \to a} f (x) + lim_{x \to a} g (x) .

Extended Sum Rule

lim_{x \to a} [f_{1} (x) + \dots + f_{n} (x)] = lim_{x \to a} f_{1} (x) + \dots + lim_{x \to a} f_{n} (x) .

Constant Function Rule

The limit of a constant function is the constant:

lim_{x \to a} C = C .

Constant Multiple Rule

The limit of a constant times a function is equal to the product of the constant and the limit of the function:

lim_{x \to a} k f (x) = k lim_{x \to a} f (x) .

Product Rule

This rule says that the limit of the product of two functions is the product of their limits (if they exist):

lim_{x \to a} [f (x) g (x)] = lim_{x \to a} f (x) \cdot lim_{x \to a} g (x) .

Extended Product Rule

Quotient Rule

The limit of quotient of two functions is the quotient of their limits, provided that the limit in the denominator function is not zero:

Power Rule

where the power can be any real number. In particular,

If then

This is a special case of the previous property.

Limit of an Exponential Function

where the base

Limit of a Logarithm of a Function

where the base

The Squeeze Theorem

Suppose that for all close to except perhaps for If

then

The idea here is that the function is squeezed between two other functions having the same limit

Solved Problems

Example 1.

Find the limit

Solution.

Example 2.