Determine the infinite limit of the following function.
Lim as x-->zero
1/x^2(x+7)
and lim as x-->3 from the left side
2/x-3
To determine the infinite limit of the function 1/x^2(x+7) as x approaches zero, we can use the concept of limits. Here's how you can solve it:
1. Start by factoring the function as much as possible. In this case, you can factor out 1/x^2 from the denominator, leaving you with (x+7).
2. Now, you can write the function as 1/(x^2) * (x+7).
3. As x approaches zero, the term 1/(x^2) becomes infinitely large, while (x+7) remains constant. Therefore, the value of the function approaches zero.
Hence, the infinite limit of the function 1/x^2(x+7) as x approaches zero is 0.
For the second question, we need to find the limit of the function 2/(x-3) as x approaches 3 from the left side. Here's how you can do it:
1. Plug in a value slightly less than 3 into the function to see if it approaches a specific value. Let's choose 2.9 as an example.
2. Evaluate the function using 2.9: 2/(2.9-3) = 2/(-0.1) = -20.
3. As you can see, as x approaches 3 from the left side, the function approaches negative infinity because the denominator becomes very close to zero from the negative side while the numerator remains constant.
Therefore, the limit of the function 2/(x-3) as x approaches 3 from the left side is negative infinity.