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Illustrations of Limits of Functions

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Illustrations of Limits of Functions - Lesson Summary

Graph of the functionf(x) = 2x + 1.

Limit of the function at two. Values of the function which are very close to two are obtained.

The values close to two are 1.999 and 2.001. There is no significant change in the value of the left hand and the right hand limits.

Both the limits are approaching five. This implies that the two limits coincide with each other and the limit of the function exists at two.

Therefore,   lim x → 2 f(x) = 5 .

The graph of the function f(x) = x + sin x

The table shows the limit of the function close to the values of 'p/2'.

The values close to ''p/2' are taken with a difference of 0.01 to the left and to the right. There is no significant change in the value of the left hand and the right hand limits.

This implies that the two limits coincide with each other and the limit of the function exists at 'p/2'.

Consider the value of the function intermediate to both the limits.

Therefore, lim n → π 2 f(x) + lim n → π 2 x + sin x  = 2.5707.

Consider a two-piece function  f ( x ) = x 2 x < 2 x 2 3 x ≥ 2 .

Graph of the function:

Values of the function which are very close to two are obtained.

The values close to two are 1.999 and 2.001. These are the points to the left and right, respectively, of the point, 2.

There is a significant shift in the value of the left hand and the right hand limits. The limit of f(x) does not exist at x = 2.

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