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Limits: Reaching for a Goal

12 Dec Posted by in Calculus | Comments

A limit can be a difficult concept to grasp in mathematics, as it is a value that is never reached, but is ‘approached’ by an input, or index of some nature. Limits are essential for integrals, continuity and derivatives in calculus.

Let f(x) be a given function. Values of f(x) can be made arbitrarily close to L. We can do this by taking x sufficiently close to a, with x≠a, on either side of a, then we say that L is the limit of f(x) as x approaches a and we write it as f(x) = L.

If xà∞ then it means that x values will go far on the right side of the horizontal axis,  x-axis. The corresponding f(x) values will be read on the vertical axis, Y-axis. To understand this we can examine the following examples.

Calculation of limits:

a)      The limit of sum of two functions is nothing but the sums of the limits of the individual functions.

[f(x)+g(x)] =  f(x) +  g(x)

b)      The limit of difference of two functions is nothing but the difference of the limits of the individual functions in the given order.

    [f(x)-g(x)] =  f(x) –  g(x)

c)      The limit of product of two functions is nothing but the product of the limits of the individual functions.

[f(x)*g(x)] =  f(x) * g(x)

d)      The limit of quotient of two functions is nothing but the quotient of the limits of the individual functions in the given order.

[f(x)/g(x)] =  f(x)/ g(x)

Situations where a limit cannot be found:

a)      The limit of a difference of functions cannot exist if both tend to ∞.

b)      The limit of product of two functions cannot exist of one tend to ∞ and the other tend to 0.

c)      The limit of quotient of two functions cannot exist if both tend to either ∞ or zero.

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