Can the two ends approach different heights?
You will be able to: Compare positive and negative end behavior and avoid sign errors.
Can the two ends approach different heights?
The ratio x/sqrt(x²+1) approaches 1 to the right and −1 to the left. The square root is always nonnegative, so its dominant term is abs(x), not x.
A useful starting point: How do limits at infinity describe end behavior? →
Words and symbols before equations
- Absolute value
- Nonnegative magnitude; sqrt(x²)=abs(x).
- End-specific limit
- Analyze x→∞ separately from x→−∞.
- Growth comparison
- Study the ratio of functions for large inputs.
- Asymptote crossing
- A finite intersection that does not alter end behavior.
What this picture assumes
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. sqrt(x²)=|x|. The exponential comparison uses positive x regardless of end selector, labeled in the readout; exponential underflow is labeled as rounding, not exact zero.
Read the picture in three steps
- Read the species and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Sample x=10, output=0.995037. Positive-end limit 1; negative-end limit −1. Use sqrt(x²)=|x|.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the chemistry
Write sqrt(x²+1)=abs(x)sqrt(1+1/x²). The ratio becomes (x/abs(x))/sqrt(1+1/x²) for x≠0.
On the positive end x/abs(x)=1, giving limit 1. On the negative end it equals −1, giving limit −1. Different horizontal asymptotes at different ends are allowed.
The curve 2+x/(x²+1) tends to 2 at both ends and equals 2 at x=0. This shows that a horizontal asymptote can be crossed.
For positive x→∞, exponential growth eˣ eventually exceeds any fixed power xⁿ, and any fixed positive power exceeds ln x. Numerical tables illustrate these standard growth comparisons; a finite plot alone is not a proof. Do not apply a positive-end statement unchanged at the negative end.
A worked example, step by step
Find lim as x→−∞ of x/sqrt(x²+1).
- Factor x² from inside the root.
- Use sqrt(x²)=abs(x), not x.
- For negative x, x/abs(x)=−1.
- The remaining square-root factor tends to 1, so the limit is −1.
The identity sqrt(x²)=x fails for negative x. Use abs(x).
Does crossing y=2 once rule out y=2 as a horizontal asymptote?
Compare with an explanation
No. A horizontal asymptote is about end behavior, not all finite inputs.
Predict. Change one thing. Explain.
Switch between the two-ended ratio, a crossing asymptote and a positive-end growth ratio. Explain the domain and sign for each comparison.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Sample x=10, output=0.995037. Positive-end limit 1; negative-end limit −1. Use sqrt(x²)=|x|.
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. sqrt(x²)=|x|. The exponential comparison uses positive x regardless of end selector, labeled in the readout; exponential underflow is labeled as rounding, not exact zero.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using nearby function values, graph coordinates, or the hypotheses of a limit law or theorem. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFind both end limits of x/sqrt(x²+4), identify the sign step, and explain why a graph alone is insufficient.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Rewrite denominator as abs(x)sqrt(1+4/x²).
- 1 point: At +∞ the ratio tends to 1.
- 1 point: At −∞ the ratio tends to −1.
- 1 point: The absolute-value identity justifies the sign; a finite window cannot show all end behavior.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What is sqrt(x²)?
abs(x).
RECALL 2Can a horizontal asymptote be crossed?
Yes.
RECALL 3Which end is assumed in ln x versus powers?
The positive end within the logarithm’s domain.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Can the two ends approach different heights?
- sqrt(x²)=abs(x).
- Analyze both input ends separately; horizontal asymptotes may be crossed.
Remember: The identity sqrt(x²)=x fails for negative x. Use abs(x).
Conditions: Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. sqrt(x²)=|x|. The exponential comparison uses positive x regardless of end selector, labeled in the readout; exponential underflow is labeled as rounding, not exact zero.
Refresh Kid · AP Calculus AB Unit 1 · Objectives LIM-2.D · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.15, LIM-2.D. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 1 has 16 official topics. Focused lesson titles and questions are original Refresh Kid teaching material. Topics 1.7 and 1.9 integrate earlier objectives and skills rather than introducing new numbered learning objectives.
Formal epsilon-delta proofs are not assessed in this unit. L’Hôpital’s rule, derivative rules and differentiation tests belong later in the course and are not used to bypass limit reasoning here. Trigonometric limits use radians. Finite graphs and tables provide evidence, not a proof of all nearby behavior. Infinity describes unbounded behavior, not a number to substitute. The Intermediate Value Theorem requires continuity on a closed interval and an intermediate output; it guarantees existence, not uniqueness.
The Organic Chemistry Tutor video title, creator and description were checked; the full video was not reviewed. Khan Academy’s current course announcement and linked unit destination were checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Section 2.2 was consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. Camera rotation does not add a variable or change slope; use labeled coordinates and axis scales.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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