Limits & Continuity

Master the formal foundation of Calculus. Explore limits $\lim_{x \to c} f(x) = L$ using the strict $\epsilon$-$\delta$ definition, identify left/right limits, and classify types of discontinuity.

Interactive Sandbox

Formal Limits & Discontinuity Graph

Drag the target value **c** directly on the graph's x-axis. Adjust epsilon (ε) and delta (δ) tolerances to evaluate the formal limit condition, or switch presets to explore holes, jumps, and vertical asymptotes.

Curve: Continuous Function
Formal Limit Condition Satisfied!
Target Value (c) 1.50
Epsilon Tolerance (ε) 0.40
Delta Neighborhood (δ) 0.20
If the curve within the vertical delta band fits entirely inside the horizontal epsilon band, the limit holds.
Algebraic Solver

Left/Right One-Sided Limit Solver

Trace one-sided limits and evaluate the three mathematical conditions required for continuity at point **c**.

One-Sided Limits & Continuity Analysis


Limit Properties Summary


Left-Hand Limit (x &to; c⁻)

lim = 2.10

Right-Hand Limit (x &to; c⁺)

lim = 2.10

Continuity Status at c

Continuous

Evaluation

Limits & Continuity Quiz

Test your knowledge of one-sided limits, continuity criteria, and classifications of discontinuity.

Question 1 of 5 Score: 0/0

If the left-hand limit is 3 and the right-hand limit is 3, what is the value of the limit?

Correct Answer!

Explanation text...

Continuity Criteria


  • Condition 1: f(c) must be defined
  • Condition 2: lim f(x) must exist (left = right)
  • Condition 3: lim f(x) = f(c)