V(x) = 350x /(1+x^2)^(3/4); x is defined in terms of 10,000 light years. What is the limiting formof V(x) when x is small (i.e.x<1) and when x is large (i.e.x>1)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The velocity V(x) = `(350*x)/(1+x^2)^(3/4)` where x is defined in terms of 10000 light years. The limiting form of V(x) has to be determined. This means the value of V(x) has to be determined for values of x as it takes on very small values or tends to 0 and as it takes on very large values or tends towards infinity.

Looking at the velocity function V(x) it is seen that as x tends to 0, the numerator 350*x tends to 0 and the denominator tends to 1. The value of V(x) therefore tends to 0.

V(x) can be written as `350/((1/x^(4/3) + x^(3/2))^(3/4))`

As x tends to infinity the value of 1/x^(4/3) tends to 0 and the value of x^(3/2) tends to infinity, the sum of the terms that form the denominator tends to infinity. If the denominator of a fraction tends to infinity, the fraction tends to 0.

V(x) tends to 0 both for values of x that tend to 0 and for values of x that tend to infinity.

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