Evaluating The Limit Of E^(6x-x^2) As X Approaches Infinity

In the realm of calculus, evaluating limits is a fundamental concept. This article delves into the process of determining the limit of the function e^(6x - x^2) as x approaches infinity. We will break down the problem into manageable steps, first analyzing the behavior of the exponent (6x - x^2) and then considering the implications for the overall exponential function.

a) Determining the Limit of the Exponent: lim (6x - x^2) as x Approaches Infinity

To begin, we must consider the limit of the exponent, which is the expression 6x - x^2, as x approaches infinity. This is a crucial step, as the behavior of the exponent will directly influence the behavior of the entire exponential function. When dealing with limits involving polynomials as x approaches infinity, the term with the highest power dominates the overall behavior. In this case, the expression 6x - x^2 contains two terms: 6x, which is a linear term, and -x^2, which is a quadratic term. As x grows infinitely large, the quadratic term -x^2 will grow much faster in magnitude than the linear term 6x. The negative sign in front of the x^2 term indicates that this term will become infinitely negative as x approaches infinity. Therefore, the limit of 6x - x^2 as x approaches infinity is negative infinity.

To demonstrate this more rigorously, we can factor out x^2 from the expression: 6x - x^2 = x^2(6/x - 1). Now, as x approaches infinity, the term 6/x approaches zero, and the expression inside the parentheses approaches -1. However, x^2 approaches positive infinity. Thus, we have the product of an infinitely large positive number (x^2) and a number approaching -1 (6/x - 1). This product will be an infinitely large negative number. Hence, we can formally express this as:

lim (x→∞) (6x - x^2) = lim (x→∞) x^2(6/x - 1) = -∞.

Understanding this limit is paramount, as it sets the stage for understanding the overall limit of the exponential function. The fact that the exponent approaches negative infinity is key to determining the final result. Remember that exponential functions with negative exponents behave differently than those with positive exponents. As the exponent becomes increasingly negative, the value of the exponential function approaches zero. This concept is essential for the next step in evaluating the limit.

b) Rewriting the Original Limit Using Substitution and Evaluating the Exponential Limit

Having established that the limit of 6x - x^2 as x approaches infinity is negative infinity, we can now rewrite the original limit using substitution. Let's introduce a new variable, t, such that t = 6x - x^2. As we determined in the previous section, as x approaches infinity, t approaches negative infinity. Therefore, we can rewrite the original limit as:

lim (x→∞) e^(6x - x^2) = lim (t→-∞) e^t

This substitution simplifies the problem significantly. We now have a standard limit involving the exponential function e^t as t approaches negative infinity. To evaluate this limit, we need to recall the behavior of the exponential function. The exponential function e^t is defined as the inverse of the natural logarithm function. It's a strictly increasing function, meaning that as the input t increases, the output e^t also increases. However, as t approaches negative infinity, e^t approaches zero. This is because a negative exponent can be interpreted as a reciprocal: e^t = 1/e^(-t). As t becomes increasingly negative, -t becomes increasingly positive, and e^(-t) becomes an infinitely large positive number. Therefore, 1/e^(-t) approaches zero.

Graphically, this can be visualized by imagining the graph of e^t. The graph starts close to the x-axis (y=0) on the left side (as t approaches negative infinity) and rises rapidly as t increases. This visual representation helps solidify the understanding that the limit of e^t as t approaches negative infinity is indeed zero. To provide a more formal explanation, we can consider the properties of exponential functions. We know that e^0 = 1, and for any negative value of t, e^t will be a positive number less than 1. As t becomes increasingly negative, e^t gets closer and closer to zero, never actually reaching it. This is the essence of a limit approaching zero.

In conclusion, the limit of e^t as t approaches negative infinity is zero. This is a fundamental property of exponential functions and a crucial concept in calculus. By substituting t = 6x - x^2, we transformed the original complex limit into a more straightforward one that we could easily evaluate. This highlights the power of substitution as a technique for solving limit problems.

Final Answer

Therefore, the final answer is:

lim (x→∞) e^(6x - x^2) = 0

This detailed explanation breaks down the problem into manageable parts, clarifying the reasoning behind each step. By understanding the behavior of the exponent and the properties of exponential functions, we can confidently determine the limit of the given expression. This approach is applicable to a wide range of limit problems involving exponential functions and polynomials.