Proving the Limit of the Integral of a Continuous Function

Consider a continuous function f defined on the semi-infinite interval [0, ∞] with the property that its limit as the argument approaches infinity is 1. Mathematically, this is expressed as limx→∞ f(x) 1. We aim to prove that the limit of the integral of f from x to x1 also approaches 1 as x approaches infinity. This can be formally stated as limx→∞ ∫??11 t dt 1.

Understanding the Given Conditions

Given that f(x) is continuous on [0, ∞] and its limit as x approaches infinity is 1, we can deduce that for any arbitrarily small positive number c, there exists a positive number N such that for all x > N, the value of f(x) is within the interval (1 - c, 1 c). In mathematical terms, we have the following inequality for large enough x:

1 - c ≤ f(x) ≤ 1 c

Integration of the Function

Now, we are interested in the integral of f from x to x1. This can be written as:

∫??11 f(t) dt

Applying the given inequality on the function f(x), we have:

1 - c ≤ f(x) ≤ 1 c

Integrating both sides of the inequality from x to x1, we get:

∫??11 (1 - c) dt ≤ ∫??11 f(t) dt ≤ ∫??11 (1 c) dt

Evaluating the Definite Integrals

Evaluating these integrals, we obtain:

(1 - c)(x1 - x) ≤ ∫??11 f(t) dt ≤ (1 c)(x1 - x)

Taking the Limit as x Approaches Infinity

Now, we take the limit of both sides of the inequality as x approaches infinity. Since (x1 - x) is a fixed quantity, its behavior as x approaches infinity is straightforward. As we are taking the limit, we can ignore the fixed quantity x1 - x:

limx→∞ (1 - c)(x1 - x) ≤ limx→∞ ∫??11 f(t) dt ≤ limx→∞ (1 c)(x1 - x)

Simplifying, we get:

) ≤ limx→∞ ∫??11 f(t) dt ≤ (1 c)

Since c is an arbitrary positive number, we can make it as small as we like. Therefore, as c approaches 0, the inequality becomes:

1 ≤ limx→∞ ∫??11 f(t) dt ≤ 1

This implies that the limit of the integral as x approaches infinity is exactly 1:

limx→∞ ∫??11 f(t) dt 1

Conclusion

We have successfully proven that for a continuous function f on [0, ∞] with limx→∞ f(x) 1, the limit of the integral of f from x to x1 as x approaches infinity is indeed 1. This result demonstrates the powerful relationship between the behavior of a function and its integral when the function's limit is known at infinity.