Is Goblins free for teachers?

Yes. Goblins offers a generous free tier that includes 1 live feedback assignment per week, insights on student mastery and replays, and the ability to upload your own content. The Max plan adds unlimited assignments, superadaptive mode, grade exports, standards growth reports, and SSO/LMS integration, with per-student pricing for schools and districts.

How does Goblins work?

Three simple steps: (1) Share an assignment from thousands of standards-aligned topics using a share code. (2) Students speak and draw their thinking naturally, and Goblins responds with Socratic questioning that sounds just like their teacher. (3) Teachers get real-time formative insights showing what students struggle with and suggestions to inform instruction.

Does Goblins prevent cheating?

Yes. Goblins uses a purpose-trained AI model that evaluates cheating signals in real time. It can detect when students are copying answers, using external tools inappropriately, or not genuinely working through problems. Copy-pasting answers is also blocked.

Can I use my own voice with Goblins?

Yes. Teachers can record their voice and take a selfie to create a custom 3D avatar in about 30 seconds. Students then hear help in their own teacher's voice and see a familiar face, making the experience more personal and effective.

What math levels does Goblins cover?

Goblins covers math from Grade 5 through AP Calculus, including thousands of standards-aligned topics. Teachers can also upload their own content for custom assignments at any level.

Is Goblins FERPA compliant?

Yes. Goblins is FERPA and state compliant. Student data privacy and security are a top priority.

Does Goblins support English Language Learners?

Yes. Goblins supports 30+ languages so teachers don't have to translate every worksheet. It is especially helpful for ELL students who need accessible math instruction in their native language.

AT.A.2.e

Use limits to describe the behavior of a function near its asymptotes and removable discontinuities.

Example Problems

Find

\lim_{x \to 0} \frac{-8}{x^3}

Find

\lim_{x \to 2} \frac{7}{x - 2}

Find

\lim_{x \to 3} \frac{-4}{2x - 5}

Khan Academy Resources

Limits intro Limits and continuity: FAQ Using tables to approximate limit values Strategy in finding limits Estimating limit values from graphs Limits using conjugates Infinite limits: algebraic Next steps after indeterminate form (finding limits)Limits at infinity of quotients with square roots Limits of trigonometric functions Limits of combined functions: sums and differences Continuity at a point (algebraic)Limits by direct substitution Strategy in finding limits Limits by factoring Infinite limits: graphical Limits of combined functions: products and quotients One-sided limits from graphs Conclusions from direct substitution (finding limits)Direct substitution with limits that don't exist Estimating limit values from graphs Limits of composite functions Continuity at a point (graphical)Limits intro Limits using trig identities Limits of piecewise functions Limits at infinity: graphical Connecting limits and graphical behavior One-sided limits from tables Limits at infinity of quotients Limits by factoring Limits of piecewise functions: absolute value Limits at infinity of quotients (Part 1)Limits at infinity of quotients (Part 2)Limits at infinity of quotients with square roots (odd power)Limits intro Introduction to infinite limits One-sided limits from graphs Limits at infinity of quotients with square roots (even power)Worked example: Continuity at a point (graphical)Functions continuous at specific x-values Infinite limits and asymptotes Limits of trigonometric functions Strategy in finding limits Connecting limits and graphical behavior Analyzing unbounded limits: rational function Introduction to limits at infinity Functions with same limit at infinity Worked example: point where a function isn't continuous Trig limit using double angle identity Limit of sin(x)/x as x approaches 0 One-sided limits from tables Limits by rationalizing Limits of composite functions: external limit doesn't exist Analyzing unbounded limits: mixed function Limit of (1-cos(x))/x as x approaches 0 Limits of composite functions: internal limit doesn't exist Theorem for limits of composite functions: when conditions aren't met Approximating limits using tables Trig limit using Pythagorean identity Limits of combined functions: piecewise functions Estimating limits from tables Estimating limit values from graphs Continuity at a point Limits of piecewise functions Limits by direct substitution One-sided limits from graphs: asymptote Undefined limits by direct substitution Theorem for limits of composite functions Types of discontinuities Unbounded limits Continuity over an interval Limits of combined functions

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