# Courses in Math. Function. Limit.

This page helps you learn how to find the limit of a function in an analytic way.

## Function Limit - Analytic Examples.

### Example 1. Find the limit of the function:

[x + [x + x1/2]1/2]1/2/[x + 1]1/2

when x tends to +∞

### limit:

lim [x + [x + x1/2]1/2]1/2/[x + 1]1/2 = lim [1 + (1/x + x-3/2)]1/2/[1 + 1/x]1/2 = 1

### Example 2. Find the limit of the function:

[x1/2 + x1/3 + x1/4]/[2x + 1]1/2

when x tends to +∞

### limit:

lim [x1/2 + x1/3 + x1/4]/[2x + 1]1/2 =
lim x1/2[1 + x-1/6 + x-1/4]/[x1/2(2 + 1/x)1/2] =
lim [1 + x-1/6 + x-1/4]/(2 + 1/x)1/2 = 1/21/2

### Example 3. Find the limit of the function:

[(1 + 2x)1/2 - 3]/[x1/2 - 2]

when x tends to 4

### limit:

lim [(1 + 2x)1/2 - 3]/(x1/2 - 2) = d'H = (1 + 2x)-1/2/(1/2 x-1/2) =
2x1/2/(2x + 1)1/2 = 4/3

### Example 4. Find the limit of the function:

[(1 - x)1/2 - 3]/[2 + x1/3]

when x tends to -8

### limit:

lim [(1 - x)1/2 - 3]/(2 + x1/3) = d'H =
[-1/2 (1 - x)-1/2]/(1/3 x-2/3) = -2

### Example 5. Find the limit of the function:

[x1/2 - a1/2 + (x - a)1/2]/(x2 - a2)1/2

when x tends to a, a > 0

### limit:

lim [x1/2 - a1/2 + (x - a)1/2]/(x2 - a2)1/2 =
lim [x1/2 - a1/2 + (x - a)1/2]/[(x - a)(x + a)]1/2 =
lim [x1/2 - a1/2]/[(x - a)(x + a)]1/2 + 1/[x + a]1/2=
lim [x - a]/[(x - a)1/2(x + a)1/2(x1/2 + a1/2)] + 1/[x + a]1/2=
lim (x - a)1/2/[(x + a)1/2(x1/2 + a1/2)] + 1/[x + a]1/2 = 1/[2a]1/2

### Sequence. Limit.

If you want to find more analytical examples how to calculate limits go to the page function - limits - analytical examples.

### Series. Convergence.

To train another analytical examples containing limits calculations go to the page series - convergence - analytical examples.