11.1 Introduction to Limits
The notion of a limit is a fundamental concept of calculus.
Create an open box from a square piece of material 24 x 24 cm on each side. Cut equal squares from each corner and turn up the sides.
(a) Draw and label a diagram that represents the box (b) Verify that the volume V of the box is given by V = 4x (12 - x)² (c) The box has a maximum volume when x = 4. Use a graphing utility to complete the table and observe the behavior of the function as x approaches 4. make a table to find limₓ₋>₄ V. |
Definition of Limit
If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, then the limit of f(x) as x approaches c is L. This is written as limx->c f(x) = L
If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, then the limit of f(x) as x approaches c is L. This is written as limx->c f(x) = L
In this example the graph is continuous. Graphs that are not continuous, it is more difficult to find the limit.
f(x) has a limit as x approaches 0 even though the function is not defined at x = 0. This often happens, and it is important to realize that the existence or nonexistence of f(x) at x = c has no bearing on the existence of the limit of f(x) as x approaches c.
Limits that Fail to Exist
Properties of Limits and Direct Substitution
Sometimes the limit can be evaluated through direct substitution. There are many well behaved functions such as polynomial functions and rational functions with non zero denominators.
Basic Limits
lim x approaches c b = b
lim x approaches c x = c
lim x approaches c xⁿ = cⁿ
lim x approaches c ⁿ√x = ⁿ√c for an even n and c > 0
Basic Limits
lim x approaches c b = b
lim x approaches c x = c
lim x approaches c xⁿ = cⁿ
lim x approaches c ⁿ√x = ⁿ√c for an even n and c > 0