Find the limit if it exists.
lim 1/(x-2) = infinity
x→2+
lim 1/(x-2) = negative infinity
x→2-
lim 1/(x-2) = Does not exist
x→2
lim (3x+2) = infinity
x→∞
lim 999/(x^3) = 0
x→-∞
Can someone check to make sure that I am correct?
all look good
Based on the provided limits, your answers are correct.
To determine the limits, we can follow a few steps:
1. Examine the behavior on the left and right sides of the point where x approaches (in this case, 2 or ∞).
- For the limit as x approaches 2 from the right side (x → 2+), we substitute a value marginally greater than 2 into the expression 1/(x-2). As the value gets closer to 2 from the right, the denominator approaches 0, and the fraction approaches infinity.
- For the limit as x approaches 2 from the left side (x → 2-), we substitute a value marginally smaller than 2 into the expression 1/(x-2). As the value gets closer to 2 from the left, the denominator approaches 0, but the fraction approaches negative infinity.
- For the limit as x approaches 2 (x → 2), we cannot substitute a specific value into the expression 1/(x-2) because the denominator is 0 at that point. As a result, the limit does not exist.
- For the limit as x approaches ∞ (x → ∞), we substitute a very large value into the expression (3x+2). As x becomes larger and larger, this expression also becomes larger and larger, approaching infinity.
- For the limit as x approaches -∞ (x → -∞), we substitute a very large negative value into the expression 999/(x^3). As x becomes more negative and approaches -∞, the fraction becomes smaller and approaches 0.
2. Based on these steps, your answers are consistent with the behavior of the given expressions.