Find the limit
f(x) = -(x^2)/(x+3)
a) lim f(x)
x-3^-
b) lim f(x)
x-3^+
c) lim f(x)
x-3
I know that a is dne because of the graph and that b is -infinity. And that c is dne because they are not the same. But I do not know how to do this algebraically. Thank you
It helps to look at the graph of
y = - x^2/(x+3)
http://www.wolframalpha.com/input/?i=plot+y+%3D+-(x%5E2)%2F(x%2B3)
look at the second graph, the first one is a close-up of the second.
The "forbidden" value is x = -3
now look at just a bit to the right of x = -3
What does the graph do ?
What does the graph do just a bit to the left of -3 ?
you might also want to use your calculator and evaluate -x^2/(x+3) when x = -3.01 and -2.99 as an example
for x = -3.01,
y = + 906.01
now try x = -2.999
y = -8994.001
use you calculator to get even closer to -3 from both sides
AB ☍ CD
AB ▒ CD
AB ∥ CD
To find the limit of a function algebraically, you can use different approaches depending on the type of function and the value you are approaching.
For the function f(x) = -(x^2)/(x+3):
a) lim f(x) as x approaches 3 from the negative side (denoted as x->3^-):
To find this limit, substitute a value slightly less than 3 into the function and observe the output. The closer the value is to 3, the more accurate the result:
lim x->3^- -(x^2)/(x+3)
Substituting x = 2.9:
f(2.9) = -(2.9^2)/(2.9+3) = -8.41/5.9 ≈ -1.4237
By further evaluating f(x) at values closer to 3 (such as 2.99, 2.999, etc.), you will notice that the output approaches closer and closer to a certain value. In this case, the limit does not exist (denoted as dne) because the function approaches different values as x approaches 3 from the left side.
b) lim f(x) as x approaches 3 from the positive side (denoted as x->3^+):
Similarly, substitute a value slightly greater than 3 into the function and observe the output:
lim x->3^+ -(x^2)/(x+3)
Substituting x = 3.1:
f(3.1) = -(3.1^2)/(3.1+3) = -9.61/6.1 ≈ -1.5754
As x approaches 3 from the right side, the function approaches a single value, in this case, -1.5754. Therefore, the limit as x approaches 3 from the positive side is -1.5754.
c) lim f(x) as x approaches 3 (denoted as x->3):
To find this limit, you need to check if the limits from both sides are equal. If they are, then the overall limit exists. However, if the limits from both sides are unequal, the overall limit does not exist.
In this case, the limit from the negative side is dne (-1.4237) and the limit from the positive side is -1.5754. Since these values are not equal, the overall limit as x approaches 3 is dne.
In summary:
a) The limit as x approaches 3 from the negative side is dne.
b) The limit as x approaches 3 from the positive side is -1.5754.
c) The overall limit as x approaches 3 does not exist.