Find the limit if it exists.
lim 1/(x-2)
x→2+
lim 1/(x-2)
x→2-
lim 1/(x-2)
x→2
lim (3x+2)
x→∞
lim 999/(x^3)
x→-∞
To find the limit of a function as x approaches a specific value or infinity/minus infinity, we can analyze the behavior of the function as x gets closer and closer to that particular value.
1. lim 1/(x-2) as x approaches 2 from the right (x→2+):
To find this limit, we evaluate the function by plugging in numbers slightly greater than 2. For example, calculate the value of 1/(x-2) for x = 2.1, 2.01, 2.001, and so on. The pattern of the values should help us determine the limit. In this case, as x approaches 2 from the right, the values of 1/(x-2) get larger and larger. Therefore, the limit is positive infinity (∞).
2. lim 1/(x-2) as x approaches 2 from the left (x→2-):
Similarly, to find this limit, we evaluate the function by plugging in numbers slightly less than 2. For example, calculate the value of 1/(x-2) for x = 1.9, 1.99, 1.999, and so on. As x approaches 2 from the left, the values of 1/(x-2) get smaller and smaller. Therefore, the limit is negative infinity (-∞).
3. lim 1/(x-2) as x approaches 2 (x→2):
This limit can be found by considering the limits from the left and right. Since the limit from the left is -∞ and from the right is +∞, the overall limit does not exist (DNE).
4. lim (3x+2) as x approaches infinity (x→∞):
To evaluate this limit, substitute a large value, such as 100, 1000, or even larger, into the function. As x approaches infinity, the value of (3x+2) also becomes infinitely large. Therefore, the limit is positive infinity (∞).
5. lim 999/(x^3) as x approaches minus infinity (x→-∞):
By substituting large negative values into the function, such as -100, -1000, or even less, we can see that as x approaches minus infinity, the value of 999/(x^3) becomes infinitely small but is still positive. Thus, the limit is 0.
Remember, when finding limits, it's important to analyze the behavior of the function as x approaches the given value or infinity/minus infinity from both sides (left and right).