Limit of a function properties theorems or laws, Squeeze rule, Composition rule, Limits of functions properties use examples

Squeeze rule

Composition rule

Limit properties

Assume that

exist and that c is any constant. Then,

11. Squeeze rule

If f(x) < g(x) < h(x) for all x in an open interval that contains a, except possibly at x = a,

and

then

12. Composition rule

If f(x) is continuous at x = b and

then,

Limits of functions properties use examples

Limits that are commonly used are written below for easy reference,

By comparing the area of the sector of the arc x with

areas of the two right triangles, the smaller triangle

with legs sin x and cos x, and the bigger triangle with

legs tan x and 1, shown on the right figure, we get

Since the left and the right side of the last inequality

tend to the same limit 1, as x tends to 0 then,

applying the squeeze rule obtained is

the fundamental or basic trigonometric limit. Note that the arc length x is measured in radians.

Further,

since both factors on the right tend to 1, as x tends to 0, then

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