Final answer:
The limit as x approaches 0 of (9^x - 2.6^2 + 4^x) / x^2 is negative infinity because as x goes to 0, both 9^x and 4^x approach 1, making the numerator negative and the denominator approach 0, thus the overall expression approaches negative infinity.
Step-by-step explanation:
The student is asking us to evaluate the limit as x approaches 0 of the function (9^x - 2.6^2 + 4^x) / x^2. To solve this, we must first recognize that as x approaches 0, the exponential functions 9^x and 4^x will approach 1, because any number raised to the power of 0 is 1. Also, the constant 2.6^2 is simply a constant and does not affect the limit as x approaches 0.
To find the limit, we treat each term separately. The limit of 9^x as x approaches 0 is 1, and the limit of 4^x as x approaches 0 is also 1. Thus, we have the function simplifying to (1 - 2.6^2 + 1) / 0, which simplifies further to (2 - 2.6^2) / 0. Since the 2.6^2 is a constant, the limit will depend on the ratio of a constant to a power of x. As x approaches 0, x^2 also approaches 0, and thus the function approaches infinity or -infinity depending on the sign of the numerator.
Since 2.6 squared is larger than 2, the numerator will be negative and we can conclude that the limit is negative infinity. Therefore, the correct answer is D) Infinity.