Pre-Calculus
Unit 1
Section 1.3 Notes – Continuity, End Behavior, and Limits
Objectives:
• Use limits to determine the continuity of a function, and apply the Intermediate Value Theorem to continuous functions.
• Use limits to describe end behavior of functions.
Continuous Function:
• no breaks, holes, or gaps
• Can be traced with a pencil without lifting your pencil
Limit:
• Approaching a value without necessarily ever reaching it
Discontinuous Function
• Functions that are not continuous
Functions with Removable Discontinuity
• Limit of f(x) at point c exists
• Value of function at c is undefined or is not the same as the value of the limit at point c.
Nonremovable Discontinuity
•
•
Infinite and jump discontinuities
Cannot be eliminated by redefining function at that point, since the function approaches different values from the left and
right at that point or doesn’t approach a single value at all and instead is increasing or decreasing indefinitely
•
If just one of the conditions for continuity is not satisfied, the function is discontinuous at x = c.
Example 1: Determine whether the given function is continuous.
a) f(x) =
1
2𝑥+1
b) f (x) = x 2 + 2x – 3
c)
Example 2: Determine whether the given function is continuous. If discontinuous, identify the type of discontinuity as infinite, jump,
or removable.
a) f(x) =
c)
1
𝑥−1
b) f(x) =
d)
𝑥−2
𝑥 2 −4
e)
If a function is continuous, you can approximate the location of its zeros by using the Intermediate Value Theorem and its corollary
The Location Principle:
Example 3:
3
a) Determine between which consecutive integers the real zeros of f(x) = x2 – x - are located on the interval [–2, 2].
4
b) Determine between which consecutive integers the real zeros of f (x) = x 3 + 2x + 5 are located on the interval [–2, 2].
End Behavior
•
•
How a function behaves at either end of the graph
What happens to the value of f(x) as x increases or decreases without bound
Example 4: Use the graph of f(x) = x 3 – x 2 – 4x + 4 to describe its end behavior.
Support the conjecture numerically.
𝑥+2
Example 5: Use the graph of f(x) = 2
to describe its end behavior.
𝑥 −𝑥−2
Support the conjecture numerically.
𝑥 2 +𝑦 2
Example 6: The symmetric energy function is E =
. If the y-value is held constant, what happens to the value of symmetric
2
energy when the x-value approaches negative infinity?