Examples for

# Limits

Limits are the underlying tool used in calculus, appearing in the definitions of continuity, derivatives and integrals. Wolfram|Alpha has the power to compute bidirectional limits, one-sided limits, supremum and infimum limits, discrete limits and multivariable limits. More information, such as plots and series expansions, is provided to enhance mathematical intuition about a limit.

### Limits

Compute limits numerically and symbolically.

#### Compute a limit:

#### Take the limit of a difference quotient:

#### Compute a limit involving abstract functions:

### Multivariable Limits

Explore limits of expressions in two or more dimensions.

#### Compute a multidimensional limit:

#### Find the value of a nested limit:

### Discrete Limits

Evaluate limiting behavior of sequences.

#### Compute discrete limits:

#### Compute discrete supremum or infimum limits:

### One-Sided Limits

Compute one-sided limits from a specified direction.

#### Compute one-sided limits at a point of discontinuity:

#### Specify the direction of approach:

### Limit Representations

Express functions in terms of limits.

#### Find limit representations for a function:

### GO FURTHER

Step-by-Step Solutions for CalculusCalculus Web AppFree Unlimited Calculus Practice Problems### RELATED EXAMPLES

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### Supremum & Infimum Limits

Find the limiting upper and lower bounds of functions.