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Calculus, all content (2017 edition)

Course: Calculus, all content (2017 edition)>Unit 1

Lesson 7: Continuity at a point

Worked example: Continuity at a point (graphical)

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.A (LO)
,
LIM‑2.A.2 (EK)
Sal gives two examples where he analyzes the conditions for continuity at a point given a function's graph.

Want to join the conversation?

• what is the specific difference between lim at x=3+ or 3- and 3 itself? and why?
• When we write 3+ or 3‾ we meant one-sided limit. 3+ means x approaches 3 from the positive (right) direction and 3‾ means x->3 from the negative (left) direction.

When we just write 3 without the + or - we are talking about two-sided limit. Mean x approach 3 from both the left and right at the same time.

For a limit (two-sided) to exist, the limit from the left must equal to the limit from the right.

So for example,

Lim x->3+ = 2 and lim x->3- = 10, since they are not the same, then lim x->3 doesn't exist.
However, if lim x->3+ = 5 and lim x->3- = 5, since they are equal, then the lim x->3 exist and is 5.

Hope that helps.
• I don't know if anyone has pointed this out yet but in the videos Sal says that for the graphs with asymptotes that no limit exists which is fair as from the left in the graphs we go through the limit is infinity which isn't a number and in the "classical" definition thus there is no limit. The problem is the questions ask the same thing in the quiz before this section but they want me to input the limit is infinity, in fact the answers available do not provide the "does not exist" option at all so Sal says one thing and the questions say another which is conflicting and confusing.
• Perhaps you're right and the question and the tutorial are inconsistent.

Know this however: you are correct, infinity is not a number, and it cannot used as the value of a limit. [It's not a number in the sense that 2 is a number, let's say]
If a function approaches infinity (either positive or negative) as the x-value approaches the asymptote (from the left or right) then the function is unbounded at that point as much as it is undefined at that point. Indeed the "limit" does not exist. However the function can reach any arbitrarily large value for any suitably chosen x value (respective of left or right) close to the asymptote. Additionally note that the value of the function exactly at the asymptote is not defined.

In rational functions, you can have asymptotes or a "removable" discontinuity. Assuming we are dealing with asymptotes, approaching the x value of the asymptote from either the left or right will result in either a positive or negative infinity. Here the limit doesn't "equal" infinity, so much that it "does not exist". In the questions, they likely only want you to stipulate whether or not a function approaches positive infinity or negative infinity as x approaches the discontinuity. Feel free to substitute Lim f(x) "approaches" for "equals".

Side-note: In future questions, I discourage you from using run-on sentences. The clarity of an answer can sometimes depend on the clarity of the question.
• what does a solid dot and a hollow dot indicate??
• A solid dot on a line indicates that the line goes through that point. A hollow dot on a line indicates that the line is not defined at that point.
• At , how come g(x) is defined when x=3+ amd x=3- are not equal?
• A function does not need to be continuous to be defined. When x = 3 there is a coordinate point that is defined that satisfies the situation, even though the limit does not exist.
• What if a function is continuous before a point, but it's not defined immediately after that point ?
• That's not uncommon. Take the function f(x)=x² on the interval [-1, 1]. f is continuous on that entire interval, including at the endpoints, but not defined past them.

You can also take this function and change the output at the points -1 and 1 only, so that the function is continuous on (-1, 1), discontinuous but still defined at -1 and 1, and undefined elsewhere.
• How do I tell whether a function is defined at a point?
• If you place the value of that point into the function, and the function solves into some real number, then the function is defined at that point.
• Couldn't the second graph be continuous AT x=3? Since the point at y=2 isn't defined, wouldn't it be possible to say that when x=3, there's only one defined point and therefore the graph is continuous at that specific point?

edit: the reason I'm confused about this is because the question doesn't ask if the graph is continuous at the limit as x approaches 3 of g(x), but whether or not the graph is continuous at x=3. I completely understand that the limit is discontinuous but i don't understand how x=3 is.
• 𝑔(2.9999...) = 2

But 2.9999... = 3, and 𝑔(3) = −2

So, even though we have chosen to define 𝑔(3) as equal to −2 there is a discontinuity at 𝑥 = 3.
• The limit exists at g(x) when approaching x from 3- side and the limit is suppose to be 2, but how can it be the limit if at 2 the point is open.
• If a certain point is not defined, but the values approaching it are, the limit still exists because when you find the limit of a certain value, you are finding the value the function approaches as x approaches that value. The function does not have to be defined at that value as long as the value of the function as x approaches that point exists (and, if you are finding a two-sided limit, the values as you approach x from both sides are equal).
• For a limit to be continuous, shouldn't the right side and the left side should be defined and also should exist without any breaks. Right?
• A limit is not continuous or discontinuous. For a limit to exist, it's necessary that the limits from left and right are both defined and equal each other.

For a function to be continuous at a point, its limit must exist at that point, and the function must equal the limit.

• If a function begins at x=1 and is continuous ahead, can we call it continuous at 1?
• Actually, we can't.

There are three conditions that must be met for a point on a graph to be continuous (I'll provide counterexamples for each condition).

1. The function must be defined at that point.
-This is straightforward. If the function is not defined at a point, it does not even exist there.
COUNTEREXAMPLE: Checking for point continuity at x=0 for a function only valid for x>5.
2. The limit of the function approaching the point in question must exist.
-The graph must connect. If the right and left-handed limits are different (or don't exist), the graph has two separate branches.
COUNTEREXAMPLE: Evaluating x=0 for f(x)={x when x>0; 5 when x<= 0}. The function is defined, but the limit does not exist.
3. The value of the defined point and the limit of the function approaching that point must have the same value.
COUNTEREXAMPLE: Evaluating at x=0 for f(x)={5 when -infinity<x<infinity; 0 when x=0}

In your case, the function you mentioned violates the 2nd and 3rd condition, as the left-handed limit cannot exist if the function begins at x=1 (If condition 2 is not met, 3 is also void).