"Lagrangian interpolation is praised for analytic utility and beauty but deplored for numerical practice." This heading, from the extended table of contents of one of the most enjoyable textbooks of numerical analysis [1],expresses a widespread view.

[...] Given (

*x0, f0*), (*x1, f1*), . . ., (*xn, fn*) with arbitrary spaced*xj*, Lagrange had the idea of multiplying each*fj*by a polynomial that is 1 at*xj*and 0 at the other*n*nodes and then taking the sum of these*n + 1*polynomials. Clearly, this gives the unique interpolation polynomial of degree*n*or less. [...]
( Erwin Kreyszig,

*Advanced Engineering Mathematics*)Figure : Lagrange Interpolation of function 1/(1+x*x) |

[...] Lagrange and other interpolation at equally spaced points, as in the
example above, yield a polynomial oscillating above and below the true
function. This behaviour tends to grow with the number of points,
leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes. [...]

( Wikipedia )

Although their are superior interpolation methods than

*lagrange interpolation*method. But it is quite easy to understand and it is superior than*Taylor Series Approximation*of a function. It can be seen easily by following diagram.
Here I choose

*in the interval [0, 2]***f(x) = e^x**
[1] F. S. Acton,

*Numerical Methods That [Usually] Work*, AMS, Providence, RI, 1990.