Limits of algebraic expressions

Maple can calculate limits with the limit(...) function. It applies many of the algorithms you have learned in calculus to find limits either as a variable goes to plus or minus infinity, or towards a specific point, again, possibly from one side or the other.

To simplify this, we will simply go through the possibilities:

$\lim_{x \rightarrow \infty} e^{-x} \sin(x)$

[> limit( exp(-x)*sin(x), x = infinity );

$0$

[> limit( exp(x)*(x^5 - 5*x + 1)*x, x = -infinity );

$0$

This requires a two-sided limit, so the limit from the left and the limit from the right mus be equal:

[> limit( (sqrt(4 + h) - 2)/h, h = 0 );

$\frac{1}{4}$

[> limit( (x^2 - 5*x + 4)/(x^2 + 3*x - 4), x = 1 );

$-\frac{3}{5}$

[> limit( 1/x, x = 0 );

$undefined$

[> limit( 1/x, x = 0, 'left' );

$-\infty$

[> limit( 1/x, x = 0, 'right' );

$\infty$

[> p := x -> x^4 - 3*x^3 + 4*x^2 + 2*x + 1:
[> limit( (p(x - h) - 2*p(x) + p(x + h))/h^2, h = 0 ); # the 2nd derivative

$12x^2 - 18x + 8$