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→-∞
Do you own a calculator?
eg
f(x) = 1/(x-2)
if x = +2.5
f(2.5) = +2
if x = 2.1
f(2.1) = +10
if x = .001
f(2.001) = +1000 etc
To find the limits, we need to evaluate the given expressions as x approaches the given values.
1. For the expression lim(1/(x-2)) as x approaches 2 from the right (x→2+), substitute x=2 into the expression:
lim(1/(x-2)) = 1/(2-2) = 1/0.
Since we get an undefined value of 1/0, the limit does not exist.
2. For the expression lim(1/(x-2)) as x approaches 2 from the left (x→2-), substitute x=2 into the expression:
lim(1/(x-2)) = 1/(2-2) = 1/0.
Again, we obtain an undefined value of 1/0, so the limit does not exist.
3. For the expression lim(1/(x-2)) as x approaches 2, we need to check if the limits from the right and left approach the same value. Since both the right and left limits are undefined (as shown in the previous steps), the overall limit does not exist.
4. For the expression lim(3x+2) as x approaches positive infinity (x→∞), substitute x with ∞:
lim(3x+2) = 3∞ + 2 = ∞.
The limit becomes infinity (∞) as x approaches positive infinity.
5. For the expression lim(999/(x^3)) as x approaches negative infinity (x→-∞), substitute x with -∞:
lim(999/(x^3)) = 999/(-∞)^3 = 999/(-∞).
Since the denominator approaches negative infinity and the numerator is a constant, the limit is 0.
Remember that these explanations show the step-by-step process to evaluate the given limits.