Example 1:

Multiplying by a unity factor

Solution:

Multiply by 1 in the form of the numerator with a "+" sign substituted for a "-" sign:

Therefore,

Please note in the above examples that, once the limit has been taken, the limit symbol is removed and the fixed point is substituted for x. Prior to that, the limit symbol is needed. When we are doing pure algebra, we leave off the limit symbol to avoid cluttering the math.

Example 2:

Example 3: Find limit:

Solution: 

Here it helps to do a substitution:

so that we can now write the limit expression as

Example 3:

Example 4: Find limit

 Solution: