given f(x) =
{ 2x^2 + 5 x < or equal to 2
3 - x^2 x > 2
find
lim f(x)
x -> 2^+
lim f(x)
x -> 2^-
lim f(x)
x -> 2
given f(x) =
{ 2x^2 + 5 x < or equal to 2
3 - x^2 x > 2
What????
Perhaps you mean:
given f(x) =
2 x^2 + 5 when x < or equal to 2
3 - x^2 when x > 2
if x is slightly bigger than 2
3 - 4 = -1
if x is slightly smaller than 2
OR
if x is 2
then
2(4) + 5 = 8+5 = 13
since the left limit does not equal the right limit, lim as x->2 does not exist.
And of course, f(2) is not even defined.
To find the limits of a function as x approaches a certain value, we need to evaluate the function as x gets arbitrarily close to that value from both the left and the right sides.
For the given function f(x) = { 2x^2 + 5, x ≤ 2; 3 - x^2, x > 2 }, let's find the limits as x approaches 2 from the right (x -> 2+) and from the left (x -> 2-).
1. lim f(x) as x approaches 2+ (x -> 2+):
To find this limit, we need to evaluate the function as x becomes closer to 2 from the right side.
Since x > 2 in this case, we'll use the second part of the function: f(x) = 3 - x^2.
Substitute x = 2 into the expression:
lim f(x) as x approaches 2+ = lim (3 - x^2) as x approaches 2+
= 3 - 2^2
= 3 - 4
= -1
Therefore, lim f(x) as x approaches 2+ (x -> 2+) is -1.
2. lim f(x) as x approaches 2- (x -> 2-):
To find this limit, we need to evaluate the function as x becomes closer to 2 from the left side.
Since x ≤ 2 in this case, we'll use the first part of the function: f(x) = 2x^2 + 5.
Substitute x = 2 into the expression:
lim f(x) as x approaches 2- = lim (2x^2 + 5) as x approaches 2-
= 2(2^2) + 5
= 8 + 5
= 13
Therefore, lim f(x) as x approaches 2- (x -> 2-) is 13.
3. lim f(x) as x approaches 2 (x -> 2):
To find this limit, we need to consider both the limits from the right and the left and see if they are equal. If they are equal, then the overall limit exists; otherwise, it does not.
Here, lim f(x) as x approaches 2 does not exist since the limit from the right (x -> 2+) is -1, and the limit from the left (x -> 2-) is 13, and they are not equal.
To summarize:
lim f(x) as x approaches 2+ (x -> 2+) = -1
lim f(x) as x approaches 2- (x -> 2-) = 13
lim f(x) as x approaches 2 (x -> 2) does not exist.