I am confused. What is the difference among the problems below? Why aren't they all undefined?
lim x->1 -2x/(1-x) = undefined
lim x->1 x/(1-x)^2 = infinity
lim x-> 3y/(4+4y+y^2) = -infinity
The differences among the limits you mentioned are due to the behavior of the expressions as they approach the specified values. Let's break it down:
1. lim x->1 -2x/(1-x) = undefined:
In this case, the expression -2x/(1-x) approaches an undefined value as x approaches 1. To understand why it is undefined, we need to simplify the expression. When you plug in x=1 into the expression, you get -2(1)/(1-1) = -2/0. Division by zero is undefined in mathematics, so the limit in this case is also undefined.
2. lim x->1 x/(1-x)^2 = infinity:
For this limit, we find that as x approaches 1, the expression x/(1-x)^2 grows larger and larger without a bound. To see this, let's examine what happens as x gets closer to 1 from the left and the right. As x approaches 1 from the left, the denominator (1-x)^2 gets smaller and smaller, resulting in a larger value for the expression. Similarly, as x approaches 1 from the right, the denominator (1-x)^2 gets smaller, and the negative sign from x/(1-x)^2 becomes positive, again resulting in a larger value. Since the expression grows infinitely larger as x approaches 1, the limit is infinity.
3. lim x-> 3y/(4+4y+y^2) = -infinity:
In this case, we have a limit with respect to both x and y. However, we are given that the limit is taken as x approaches a specific value, not y. Therefore, y is treated as a constant here. As x approaches the given value, the expression 3y/(4+4y+y^2) approaches negative infinity. To determine this, we consider the behavior of the expression as x gets very close to the given value. As x approaches the specified value, the expression becomes 3y/(4+4y+y^2). The denominator, 4+4y+y^2, approaches a non-zero value as x gets closer to the specified value. However, since the numerator, 3y, remains constant, and the denominator approaches a positive value, the entire expression becomes very negative, approaching negative infinity.
In summary, the limits you mentioned have different behaviors due to the specific expressions involved and the values they approach. It is important to carefully examine the terms and evaluate the limits to understand their individual outcomes.