limit

Calculate the bidirectional limit of this symbolic expression as x approaches 0.

syms x h
f = sin(x)/x;
limit(f,x,0)

ans = $$1$$

Calculate the limit of this expression as h approaches 0.

f = (sin(x+h)-sin(x))/h;
limit(f,h,0)

ans = $$\cos \left(x\right)$$

Calculate the right- and left-side limits of symbolic expressions.

syms x
f = 1/x;
limit(f,x,0,"right")

ans = $$\infty$$

limit(f,x,0,"left")

ans = $$-\infty$$

Because the limit from the left is not equal to the limit from the right, the two-sided limit does not exist. In this case, limit returns NaN (Not a Number).

limit(f,x,0)

ans = $$\mathrm{NaN}$$

Calculate the limit of expressions in a symbolic vector. limit acts element-wise on the vector.

syms x a
V = [(1+a/x)^x exp(-x)];
limit(V,x,Inf)

ans = $$\left(\begin{array}{cc}{\mathrm{e}}^a& 0\end{array}\right)$$

Calculate the right-side limit of ${\mathit{x}}^{\mathit{n}}$ as $\mathit{x}\to 0^{+}$ for real $$n$$ and then for positive $$n$$.

Create the symbolic variables x and n. When you create symbolic variables, they are assumed to be complex by default. Set the assumption that n is real.

syms x n
assume(n,"real")

Find the limit using this assumption.

limit(x^n,x,0,"right")

ans = 
$$\{\begin{array}{cl}1& \text{ if }n=0\\ 0& \text{ if }0<n\\ \infty & \text{ if }n<0\end{array}$$

Next, assume that n is positive. Find the limit using this assumption.

assume(n>0)
limit(x^n,x,0,"right")

ans = $$0$$