Limits
Evaluate limits of functions and sequences, resolve indeterminate forms and test continuity.
Problems
- A Value Forced by Continuity
- Continuity of a Piecewise Function
- A Product That Has to Stay Continuous
- How Fast Does It Vanish?
- A Limit Function Made Continuous
- Limit of n² Times a Root Difference
- Constants from a Sequence Limit
- Difference of Two Square Roots
- n(a^(1/n) − 1)
- n(e^(a/n) − e^(b/n))
- n²(1 − cos(π/n))
- nth Root of 1 + 2ⁿ + 3ⁿ
- √(n+1) − √n
- A Ratio of Quadratics
- A Large Constant Does Not Matter
- aⁿ/(2 + aⁿ) for |a| > 1
- √(n² + 4n + 5) − (n − 1)
- √n(√(n+2) − √(n+1))
- Sum minus Half of n²
- a²/n³ times a Sum of Squares
- Σ 1/(k(k+1))
- Σ 1/((2k−1)(2k+1))
- (5·3ⁿ + 3(−2)ⁿ)/3ⁿ
- √(aₙ² + n) − aₙ
- Difference of Roots of Arithmetic Sums
- n-th Root of the Central Binomial Coefficient
- Limit from a Weighted Sum of Squares
- A Recurrence with a Hidden Invariant
- (1/n + e^(1/n))ⁿ
- A Telescoping Product of 1 − 1/k²
- A Quartic Root minus a Product
- 1 + 2q + 3q² + ⋯
- Σ (2k − 1)/2ᵏ
- Two Bases, the Larger Wins
- A Function Defined by a Limit I
- A Function Defined by a Series
- Σ k/(k + 1)!
- A Function Defined by a Limit II
- A Function Defined by a Series II
- A Walk That Alternates Direction
- When Two Limits Agree
- nth Roots of 2 and n!
- A Sequence Defined by Factorial Sums
- A Ratio of Root Differences
- Three Consecutive Factors
- Partial Sums Multiplying to a Power of Two
- Radical Limit with Unknown Constants
- Limit of Logarithmic Hyperbolic Cosine
- x − arcsin x over x² sin 3x
- Limit of Rational Power
- Constants from an Exponential–Cosine Limit
- Constants from a Trigonometric–Log Limit
- Matching Orders in a Limit
- sin(sin x) − x over x³
- ln(x/sin x) over x²
- xˣ − x near x = 1
- 1 − cos t² up to √x
- Recovering f from a Limit
- Limit of Arcsine minus Arctangent
- Three Diverging Terms That Balance
- Taylor Expansions to Order 4
- An Integral over a Root Difference
- Limit of an Integral over x³
- Limit of an Integral of an Absolute Value
- Limit of an Integral with a Logarithm
- Harmonic Sum from n+1 to 6n
- An Averaged Integral Near Zero
- Limit of an n-th Root of a Product
- Differentiating an Integral with x Inside
- n-th Root of (2n)! over (2n)²ⁿ
- Ratio of Two Root Sums
- Growth of ∫e^(t²)
- Squared Integral over an Integral
- An Integral to sin 2x over 1 − cos x
- A Substitution Inside a Limit
- t ln t from cos x to 1
- A Ratio of Two Convolutions
- A Shrinking Integral, Magnified
- Limit of a Product Raised to a Vanishing Power
- Logarithms of Binomial Coefficients
- An Average Over a Vanishing Interval
- Limit of a Logarithmic Ratio
- Constants from a Sine-of-Sine Limit
- A Limit as a Derivative
- Limit of the Log of an Exponential Average
- Limit of a Sum of Floor Functions
- Limit of a Ratio of Power Sums
- A Polynomial from Derivative Limits
- A Trigonometric 0/0 Limit
- A Polynomial from Two Limits
- Sum of Fifth Powers over n⁶
- A Quadratic from Reciprocal Limits
- A Basic Radical Limit
- Limit of 1 − cos²(3x)
- Constants from a Quadratic Limit
- A Finite Limit at Infinity
- The Largest Base Wins
- Tuning Two Square Roots
- cos x to the Power 1/x²
- Sum of Squares over a Quadratic
- x − x² ln(1 + 1/x)
- An Equivalent Infinitesimal
- (x + √(1 + x²))^(1/x) at Infinity
- Roots of Consecutive Triangular Numbers
- (1/x)(1/sin x − 1/tan x)
- (1 − x) tan(πx/2)
- 1/x − 1/ln(1 + x)
- Bounded Oscillations at Infinity
- Sum over Difference of Sines
- ln(1 + 2ˣ)·ln(1 + 3/x)
- (1 + 1/x − 1/x²)ˣ
- nth Root of an Exponential Sum
- A Polynomial from Two Limits III
- A Quadratic from a Squared Limit
- Three Limits from a Graph
- Functional Equation with two variables I
- Cubic Polynomial Limit
- Polynomial Limit I
- Polynomial from a Limit at Infinity
- Polynomial from Two Limits
- Limit with the Inverse of ln(tan x)
- Polynomial from Limits at Zero and One
- Root Gap from a Limit Ratio
- Maximising f(2) under a Constraint
- Constants from a Limit at x = 1
- Limit from a Sum of Matrix Powers
- A Cube-Root Limit and a Double Root
- A Cubic from Two Root Limits
- A One-Sided Limit with the Floor Function
- Cancelling a Square Root at Infinity
- An Odd Cubic from Two Limits
- Growth of an Inverse Function
- When f and Its Inverse Meet
- (√(x²+1) + x)^(1/x) near 0
- A Product of Two Logarithms
- x²(a^(1/x) − a^(1/(x+1)))
- tan 2x · tan(π/4 − x)
- (1 + x sin x)ˣ near 0
- (cos x)^(sin x) near 0
- Constants from a Root-Difference Limit
- A Limit at Minus Infinity
- √(1 + tan x) − √(1 + sin x)
- Constants from sin x/(eˣ − a)
- A Convolution over eˣ² − cos x
- (cot x)^(sin 3x)
- ((2/π) arctan x)^(x²)
- Both Ends of ∞ with arctan
- An Infinite Product of Cosines
- ∛(x³ + 2x² + 1) − x
- (n sin(1/n))^(n²)
- Difference of Two Cube Roots
- Recovering a Limit of f
- Four Limits of a Quartic
- Reading One-Sided Limits from a Graph I
- Reading One-Sided Limits from a Graph III
- What Convergence Says About the Terms
- The Gap Between Two Exponentials
- Two Sequences From One Sine
- Matching the Order of Vanishing
- Two Limits With Absolute Values
- A Double-Factorial Limit
- A Polynomial from a Reciprocal Limit
- When a Floor-Function Limit Exists
- A Cubic and a Quadratic from Limits
- x²[((x+1)/(x−1))^(1/x) − 1]
- A 1^∞ Limit with an Integral
- An Exponentially Weighted Average
- Polynomials with Natural Coefficients
- Square Roots Near x = 1
- A Squeeze Between Two Sums
- Reading One-Sided Limits from a Graph II
- Two-Digit Numbers a Sine Accepts
- Three Areas as an Angle Vanishes
- Where a Root Curve Meets the Tangent
- A Vertex Roaming on a Circle
- A Quadratic and a Cubic