Evaluate .

Solution: By the properties of limits,

.

Example 2

Evaluate .

Solution: This indeterminate form can be reduced to the form , which is easily evaluated:

Example 3

Find .

Solution: We cannot substitute zero for x, since the expression under the sign of the limit is an indeterminate form . It is necessary to transform the expression and reduce the common infinitesimal factors:

.

Example 4

Evaluate .

Solution: Using the same idea as above we obtain

Find

Solution: In order to evaluate the indeterminate form , divide the numerator and denominator of the given expression by x² and then apply the properties of limits:

Short solution: Infinite function is equivalent to as , while . Therefore,

.

Example 6

Evaluate .

Solution: To evaluate the indeterminate form , multiply the numerator and denominator by factor to compete the difference between two squares. Then reduce the common infinitesimal factors:

Example 7

Evaluate .

Solution: Likewise the above, we need to reduce the common infinitesimal factor. Note that

and

.

Therefore,

.

Example 8

Evaluate .

Solution: Using the formula of the difference between two cubes, factor the numerator and reduce the common multipliers of the numerator and denominator:

.