One-sided limits from graphs: asymptote | Limits and continuity | AP Calculus AB | Khan Academy

1. Formula
   Observation

   The function is labeled as y = g(x) on the graph.

g(x)

The dependent variable representing the output of the function g for a given input x.

Displayed real-input branches; a full algebraic domain is unspecified.

1. Formula
   Observation

   The independent variable is represented by x on the horizontal axis.

x

The independent variable representing the input to the function g.

Real numbers

1. Formula
   Observation

   The limit notation is written as lim_{x -> 6^-} g(x).

lim

The mathematical operator indicating the limit of a function as the input approaches a specific value.

1. Formula
   Observation

   The symbol ∞ is used to denote that the function's value becomes unbounded.

∞

Represents an unbounded increase in the value of the function; not a specific number.

1. Formula
   Observation

   The function is labeled y = g(x) on the graph.

g(x)

The function whose graph is shown.

The displayed domain branches approach x=6 from each side; a full algebraic domain is not provided.

1. Formula
   Observation

   The expression \lim_{x \to 6^-} g(x) is written on the screen.

\lim_{x \to 6^-} g(x)

The left-hand limit of g(x) as x approaches 6.

x < 6

1. Formula
   Observation

   The expression \lim_{x \to 6^+} g(x) is written on the screen.

\lim_{x \to 6^+} g(x)

The right-hand limit of g(x) as x approaches 6.

x > 6

1. Formula
   Observation

   The text 'not exist' is written in pink next to the left-hand limit expression.

not exist

Indicates that the limit does not have a finite value.

1. Formula
   Observation

   The number -3 is written in green next to the right-hand limit expression.

-3

The finite value that the function approaches from the right side.

1. Audio
   Observation

   The explanation specifies approach toward the target from the left side.

2. Formula
   Observation

   The expression lim_{x -> 6^-} g(x) is written on the screen.

A one-sided limit describes the behavior of a function as its input approaches a specific value from only one direction (either from values less than or greater than the target). In this case, it is the limit as x approaches 6 from the left.

1. x must approach 6 from values strictly less than 6.

1. Audio
   Observation

   The explanation follows nearby plotted inputs and the rising branch near the vertical asymptote.

2. Diagram
   Observation

   The graph shows a dashed vertical line at x=6, with the blue curve rising steeply towards positive infinity as it gets closer to the line from the left.

A vertical line x=c is a vertical asymptote when the function tends to positive or negative infinity from at least one side. Mere oscillatory unboundedness need not be a definite infinite limit.

1. At least one one-sided limit is positive or negative infinity; the sign is definite.

1. One-Sided Limit

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

2. Formula
   Observation

   The expressions \lim_{x \to 6^-} g(x) and \lim_{x \to 6^+} g(x) are used.

A one-sided limit describes the behavior of a function as the input variable approaches a specific point from only one direction (either from values less than the point or greater than the point).

1. The domain contains points arbitrarily close to the target from the specified side.

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

2. Animation
   Observation

   The graph shows the function increasing without bound as x approaches 6 from the left.

The left branch shown grows without bound as x approaches6 from below, so it has no finite real limit. Under the extended infinite-limit convention this particular upward divergence is written as a limit of +infinity. General unbounded behavior need not have a definite infinite-limit sign.

1. The stated nonexistence concerns a finite real limit.

2. A limit of positive infinity describes eventual growth beyond every positive bound; infinity is not a finite real number.

1. One-Sided Limit

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

In the context of standard limits, infinity (∞) is used as descriptive notation for unbounded behavior and does not represent a specific numerical value.

1. We are working within the standard definition of limits over real numbers.

Universal

1. Audio
   Observation

   The explanation follows nearby plotted inputs and the rising branch near the vertical asymptote.

2. Animation
   Observation

   Pink dots appear sequentially on the graph at x=2, 3, 4, 5, 5.5, and 5.75, with horizontal dashed lines projecting their y-values to the y-axis.

1. Expression

   $$g(2) \approx 1.2$$
   Explanation

   Selecting x=2, which is less than 6, and observing the graph shows the y-value is slightly above 1.

   Justification

   Graphical reading

   Shown in the video
2. Expression

   $$g(3) \approx 1.5$$
   Explanation

   Selecting x=3, the y-value increases further.

   Justification

   Graphical reading

   Shown in the video
3. Expression

   $$g(4) \approx 1.8$$
   Explanation

   Selecting x=4, the y-value continues to increase, remaining under 2.

   Justification

   Graphical reading

   Shown in the video
4. Expression

   $$g(5) \approx 3$$
   Explanation

   Selecting x=5, the y-value jumps to approximately 3.

   Justification

   Graphical reading

   Shown in the video
5. Expression

   $$g(5.5) \approx 5$$
   Explanation

   Selecting x=5.5, the y-value increases to around 5.

   Justification

   Graphical reading

   Shown in the video
6. Expression

   $$g(5.75) \approx 9$$
   Explanation

   Selecting x=5.75, very close to 6, the y-value shoots up to around 9.

   Justification

   Graphical reading

   Shown in the video
7. Expression

   $$\lim_{x \to 6^-} g(x) = \infty$$
   Explanation

   As x gets closer and closer to 6 from the left, the y-values grow without bound, leading to the conclusion that the limit is infinity.

   Justification

   Definition of infinite limits

   Shown in the video

The displayed branch tends to positive infinity from the left of6. Approximate samples illustrate this plotted behavior; no algebraic function formula or general proof is supplied.

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

2. Animation
   Observation

   The cursor traces the blue curve upwards towards positive infinity as it nears the vertical dashed line at x=6 from the left.

1. Explanation

   Observe the graph of g(x) for x-values less than 6 as they get closer to 6.

   Justification

   Definition of a left-hand limit.

   Shown in the video
2. Explanation

   Notice that the y-values of the function increase rapidly and do not settle on any specific finite number.

   Justification

   Visual inspection of the graph's behavior near the vertical asymptote.

   Shown in the video
3. Explanation

   Conclude that because the function is unbounded, the left-hand limit does not exist.

   Justification

   Standard convention for limits exhibiting infinite behavior when seeking finite values.

   Shown in the video

\lim_{x \to 6^-} g(x) = \text{not exist}

1. Audio
   Observation

   The explanation traces the right branch toward the open circle and concludes its finite limit.

2. Animation
   Observation

   The cursor moves along the lower part of the graph from right to left, getting closer to the open circle at (6, -3).

1. Explanation

   Consider x-values greater than 6, such as x=8, x=7, x=6.5, x=6.01, and x=6.0000001.

   Justification

   These values represent approaching 6 from the right side.

   Shown in the video
2. Explanation

   Observe the corresponding y-values on the graph for these x-inputs.

   Justification

   Reading the function's output from its graphical representation.

   Shown in the video
3. Explanation

   For inputs sufficiently close to6 from the right, function values approach-3. The listed farther sample points need not approach that height monotonically.

   Justification

   Visual trend of the curve approaching the horizontal level y=-3.

   Supplementary explanation
4. Explanation

   Conclude that the right-hand limit is -3.

   Justification

   Definition of a limit based on the function's approach to a specific value.

   Shown in the video

\lim_{x \to 6^+} g(x) = -3

1. Audio
   Observation

   The explanation specifies approach toward the target from the left side.

2. Diagram
   Observation

   The graph of y=g(x) is displayed, showing the relevant section near x=6.

Determine the value of \lim_{x \to 6^-} g(x) given the graph of y = g(x).

1. The graph of the function y = g(x).

2. A vertical dashed line at x = 6 indicating an asymptote.

3. The curve approaches the asymptote from the left, going upwards.

Evaluate the left-hand limit of g(x) as x approaches 6.

1. Expression

   $$\text{Observe } x \to 6^-$$
   Explanation

   Identify that we need to look at the behavior of the graph for x-values strictly less than 6, moving towards 6.

   Justification

   Definition of left-hand limit

   Shown in the video
2. Expression

   $$\text{Trace the curve upwards}$$
   Explanation

   Follow the blue curve from left to right as it gets closer to the vertical line at x=6. The y-values are increasing rapidly.

   Justification

   Visual inspection of the graph

   Shown in the video
3. Expression

   $$\lim_{x \to 6^-} g(x) = \infty$$
   Explanation

   Since the y-values increase without bound as x approaches 6 from the left, the limit is described as infinity.

   Justification

   Definition of infinite limits

   Shown in the video

\infty

The graphical evidence clearly shows the function growing unboundedly large as it nears the vertical asymptote from the left side.

1. Diagram
   Observation

   The graph has a left-side upward vertical asymptote at x=6 and a separate right branch approaching(6,-3).

2. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

Determine \lim_{x \to 6^-} g(x) and \lim_{x \to 6^+} g(x) given the graph of y=g(x).

1. The graph of the function y=g(x).

2. A vertical dashed line at x=6 indicating an asymptote or discontinuity.

3. The upper branch of the graph goes to +\infty as x \to 6^-.

4. The lower branch of the graph approaches the point (6, -3) as x \to 6^+.

Evaluate the left-hand and right-hand limits of g(x) at x=6.

1. Explanation

   For the left-hand limit, observe the graph as x approaches 6 from values less than 6. The curve shoots upwards indefinitely.

   Justification

   Graphical analysis of the function's behavior to the left of x=6.

   Shown in the video
2. Expression

   $$\lim_{x \to 6^-} g(x) = \text{not exist}$$
   Explanation

   Since the function is unbounded, the limit does not exist as a finite number.

   Justification

   Definition of non-existent limits due to infinite behavior.

   Shown in the video
3. Explanation

   For inputs sufficiently close to6 from the right, function values approach-3. The listed farther sample points need not approach that height monotonically.

   Justification

   Numerical and graphical analysis of the function's behavior to the right of x=6.

   Supplementary explanation
4. Expression

   $$\lim_{x \to 6^+} g(x) = -3$$
   Explanation

   The function approaches the finite value -3 from the right side.

   Justification

   Definition of a limit.

   Shown in the video

\lim_{x \to 6^-} g(x) = \text{not exist}, \quad \lim_{x \to 6^+} g(x) = -3

The results match the visual features of the graph: an upward vertical asymptote on the left and a curve ending at an open circle at y=-3 on the right.

1. Diagram
   Observation

   A coordinate plane with a grid, axes labeled from -9 to 9, and a blue curve representing y=g(x). A vertical dashed line is at x=6.

1. Coordinate axes

2. Grid lines

3. Blue curve y=g(x)

4. Vertical dashed line at x=6

5. Open circle at (6, -3)

1. The shape and position of the graph of g(x)

2. The location of the vertical asymptote at x=6

Provides the visual data necessary to evaluate the limit of the function.

1. Animation
   Observation

   Pink dots appear on the blue curve at x=2, 3, 4, 5, 5.5, and 5.75. Horizontal pink dashed lines project these points to the y-axis, showing their approximate y-values.

1. Pink dots on the curve

2. Horizontal pink dashed projection lines

1. Pink dots appear one by one from left to right along the curve as x approaches 6.

2. Corresponding horizontal lines show the increasing y-values on the y-axis.

1. The underlying graph of g(x) remains unchanged.

Demonstrates numerically how the function's output grows as the input approaches 6 from the left.

1. Animation
   Observation

   The text 'lim_{x -> 6^-} g(x) =' is handwritten in teal ink on the right side of the screen.

1. Handwritten text

1. The limit expression is written step-by-step.

Formalizes the question being asked about the graph's behavior.

1. Animation
   Observation

   The infinity symbol '∞' is handwritten in pink ink next to the limit expression.

1. Handwritten infinity symbol

1. The symbol appears to complete the equation.

States the conclusion of the limit evaluation based on the graphical evidence.

1. Animation
   Observation

   The mouse cursor moves up the blue curve on the left side of the vertical asymptote at x=6.

1. Blue curve of g(x)

2. Vertical dashed line at x=6

3. Mouse cursor

1. Cursor position moves upwards along the curve.

2. Y-values of the function increase towards positive infinity.

1. X-values remain less than 6 but get closer to 6.

Demonstrates the unbounded behavior of the function as x approaches 6 from the left.

1. Animation
   Observation

   The mouse cursor moves leftwards along the lower blue curve, starting from around x=8 and stopping near x=6.

1. Lower blue curve of g(x)

2. Open circle at (6, -3)

3. Mouse cursor

1. Cursor position moves leftwards towards x=6.

2. Function values approach -3 as the input tends to6 from the right; they need not move monotonically throughout the earlier traced branch.

1. X-values remain greater than 6 but get closer to 6.

Demonstrates the function approaching the finite value -3 as x approaches 6 from the right.

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

Believing that writing a limit equals infinity means the limit evaluates to a specific, finite numerical value.

Infinity is a descriptive term used to indicate that the function grows without bound; it is not a real number.

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

Confusing an infinite-limit convention with the existence of a finite real limit.

The source exercise writes not exist for this unbounded branch because it seeks a finite real limit. It is also valid to describe the specific upward divergence using +infinity in extended infinite-limit notation; these conventions are not contradictory.

1. Audio
   Observation

   The explanation follows nearby plotted inputs and the rising branch near the vertical asymptote.

2. Diagram
   Observation

   The graph visually links the behavior near the vertical line x=6 to the function's output.

Evaluating a one-sided limit that results in infinity confirms the presence of a vertical asymptote at that x-value.

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

The method of evaluating one-sided limits is applied to determine that the left-hand limit does not exist due to unbounded behavior.

1. Audio
   Observation

   The explanation follows nearby plotted inputs and the rising branch near the vertical asymptote.

1. One-Sided Limit
2. Evaluating the one-sided limit using sample points

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

1. One-Sided Limit
2. Infinity is not a real number
3. Confusing infinity with a real number

1. Audio
   Observation

   The narration distinguishes upward divergence from the existence of a finite real limit.

2. Formula
   Observation

   \lim_{x \to 6^-} g(x)

1. One-Sided Limit
2. Unbounded Behavior and Non-Existent Limits
3. Evaluating the Left-Hand Limit Graphically

1. Audio
   Observation

   The explanation traces the right branch toward the open circle and concludes its finite limit.

2. Formula
   Observation

   \lim_{x \to 6^+} g(x) = -3

1. One-Sided Limit
2. Evaluating the Right-Hand Limit Graphically