PhizKid said:
The limit of the function as x approaches infinity is 3. This is because as x gets larger and larger, the term x^2 becomes much larger than the sine term and therefore the function behaves like 3/x^2 which approaches 0 as x approaches infinity.
The limit of the function as x approaches 0 is 1. This is because as x approaches 0, the term sinx becomes negligible compared to x^2 and the function behaves like x^2/x^2 which simplifies to 1.
Yes, the function is continuous at x=0. This is because the limit of the function as x approaches 0 is equal to the value of the function at x=0, which is 1. Therefore, there are no "jumps" or breaks in the graph of the function at x=0 and it is continuous.
The function approaches positive infinity as x approaches values close to pi. This is because as x gets closer to pi, the sine term approaches 1 and the numerator becomes much larger than the denominator, causing the function to approach positive infinity.
The function approaches negative infinity as x approaches values close to -pi. This is because as x gets closer to -pi, the sine term approaches -1 and the numerator becomes much larger than the denominator, causing the function to approach negative infinity.