Find the limit at a point:

Find the limit at a symbolic point:

Find the limit at -Infinity:

The nested limit as first and then :

The nested limit as and then :

Compute the multivariate limit as :

Use lim to enter the  character, and to create an underscript:

Take a limit from above or below by using a superscript or on the limit point:

After typing zero, use to create a superscript:

To specify a direction of Reals or Complexes, enter the domain as an underscript on the  character:

Enter the rule as ->, use to create an underscript, and type reals to enter :

TraditionalForm formatting:

Polynomial and rational functions:

Algebraic functions:

Trigonometric functions:

A trigonometric function with a vertical asymptote:

A wildly oscillatory function with no limit at the origin:

Exponential functions:

Logarithmic functions:

Functions like Sqrt and Log have a two-sided limit along the negative reals:

If approached from above in the complex plane, the same limit value is reached:

However, approaching from below in the complex plane produces a different limit value:

This is due to branch cut where the imaginary part reverses sign as the axis is crossed:

The limit in the complex plane does not exist:

Limits of algebraic functions at ±Infinity:

Limits of trigonometric functions at ±Infinity:

Limits of exponential and logarithmic functions at infinity:

Compute nested exponential-logarithmic limits:

A discontinuous piecewise function:

A left-continuous piecewise function:

UnitStep is effectively a right-continuous piecewise function:

RealSign is effectively a discontinuous piecewise function:

Find the limit of Floor as x approaches from larger numbers:

Find the limit of Floor as x approaches from smaller numbers:

Limits involving Gamma:

Limits involving Bessel-type functions:

Limits involving exponential integrals:

At every non-positive even integer, Gamma approaches from the left and from the right:

The signs are reversed at the negative odd integers:

Compute the nested limit as first and then :

The same result is obtained by computing two Limit expressions:

Computing the limit as first and then yields a different answer:

This is again equivalent to two nested limits:

The nested limit as first and then is :

The nested limit as first and then is :

Consider the function for two variables at the origin:

The iterated limit as and then is :

The iterated limit as and then is :

Since the value of the limit depends on the order, the bivariate limit does not exist:

Visualize the function and the values along the two axes computed previously:

Compute the bivariate limit of a function as :

The limit value is if for all , there is a where implies :

For this function, the value will suffice:

The function lies in between the two cones with slope :

Find the limit of a multivariate function:

Various iterated limits exist:

Approaching the origin along the curve yields a third result:

The true two-dimensional limit of the function does not exist:

Visualize the limit near the origin:

Find the limit of a bivariate function at the origin:

The true two-dimensional limit at the origin is zero:

Re-express the function in terms of polar coordinates:

The polar expression is bounded, and the limit as is:

Compute the limit of a trivariate function:

The limit at the origin does not exist:

Limits in the and planes do exist:

But the limit in the plane is direction dependent:

Visualize the function:

Specify conditions on parameters using Assumptions:

Different assumptions can produce different results:

Parameter‐dependent limit:

Limit from below:

Equivalently:

Limit from above:

Equivalently:

Limits at piecewise discontinuities:

Limits at a simple pole:

Limits at a branch cut:

Compute the bivariate limit approach from different quadrants:

Approaching the origin from the first quadrant:

Equivalently:

Approaching the origin from the second quadrant:

Approaching the origin from the left half-plane:

Approaching the origin from the bottom half-plane:

Visualize the function:

Return a result without stating conditions:

This result is only valid if n>0:

Return unevaluated if the results depend on the value of parameters:

By default, conditions are generated that return a unique result:

By default, conditions are not generated if only special values invalidate the result:

With GenerateConditions->True, even these non-generic conditions are reported:

Use Method{"AllowIndeterminateOutput"False} to avoid Indeterminate results:

For oscillatory functions, bounds will be returned as Interval objects:

Use Method{"AllowIntervalOutput"False} to avoid Interval object results:

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

The limit of the function as is :

This means that for values of close to , has a value close to :

Visualize the function:

The limit makes no statement about the value of at , which in this case is indeterminate:

The function does not have a limit as approaches :

The function has the limit zero:

In increasingly small regions around , continually bounces between , but gets increasingly flat:

The following rational function has a finite limit as :

Compute the that ensures that whenever :

The complicated result can be simplified by focusing on in the range between 0 and 2:

The plot of always "leaves" the rectangle of height and width centered through the sides, not the top or bottom:

Find vertical and horizontal asymptotes of a rational function:

Compute where the denominator is zero:

Verify that the function approaches at the computed values:

Compute the limit value at :

Visualize the function and its asymptotes:

Find the non-vertical, linear asymptote of a function:

Slope of the asymptote:

Compute the asymptote's vertical intercept:

Visualize the function and its asymptote:

Classify the continuity or discontinuity of f at the origin:

It is not defined at 0, so it cannot be continuous:

Moreover, the limit as x0 does not exist:

However, the limit from below exists:

The limit from above also exists, but has a different value:

Therefore, f has a jump discontinuity at 0:

Classify the continuity or discontinuity of f at the origin:

The function is defined:

The two-sided limit exists but does not equal the function value, so this is a removable discontinuity:

Find and classify the discontinuities of a piecewise function:

The function is not defined at zero so it cannot be continuous there:

The function tends to Infinity (on both sides), so this is an infinite discontinuity:

Next find where the limit does not equal the function:

The limit does exist at x==3, so this is a removable discontinuity:

The function is discontinuous at every multiple of :

For example, at the origin it gives rise to the indeterminate form :

At every even multiple of , the two-sided limit of exists:

This is also true at the odd multiples of , with a different limit:

However, at the half-integer multiples of , the two sided limit does not exist:

The function agrees with where both are defined, but it is also continuous at multiples of :

Show that is continuous at the origin:

Visualize and :

Determine whether the following function is continuous at the origin, and whether limits along rays exist:

The bivariate limit does not exist, so the function is not continuous:

The limit in the left half-plane exists and is zero, so any ray approaching from there has the same limit:

Approaching along the line with gives a result in terms of the slope:

Visualize the four rays with slope :

Compute the derivative of using the definition of derivative:

First compute the difference quotient:

The derivative is the limit as of the difference quotient:

Compute the derivative of at :

The limit of the difference quotient does not exist, so is not differentiable at the origin:

Note that the left and right limits of the difference quotient exist but are unequal:

In this case, the left and right derivatives equal the limits of from the left and right:

Visualize and its derivative; the former has a "kink" at zero, the latter a jump discontinuity:

Determine the differentiability of at :

The limit of the difference quotient exists, so is differentiable and :

Note that the limit as of does not exist, so is discontinuous:

Determine the differentiability of at the point :

The partial derivative with respect to x exists:

As does the one with respect to y:

However, the linearization condition fails, so is not differentiable:

Visualize the function:

Note that the partial derivatives of exist everywhere:

But they are discontinuous at the point :

The derivative is defined as the limit of the difference quotient:

The second derivative can be computed by taking the limit of the second-order difference quotient:

Directly compute the mixed partial derivative by taking a limit: