Use a table to frind the indicated limit.
3
7. lim(4x )
x-2
2
8. lim (2x + 1)
x-3
x + 1
9. lim 2
x-0 x + 1
To find the indicated limits using a table, we can substitute different values of x that approach the given point.
For problem 7:
We substitute x values that approach 2:
When x is slightly less than 2 (x = 1.9), the expression becomes: (4(1.9))/(1.9 - 2) = 7.6/-0.1 = -76
When x is slightly greater than 2 (x = 2.1), the expression becomes: (4(2.1))/(2.1 - 2) = 8.4/0.1 = 84
As x gets closer to 2, the expression approaches either positive or negative infinity.
Therefore, the limit as x approaches 2 of (4x)/(x-2) does not exist.
For problem 8:
We substitute x values that approach 3:
When x is slightly less than 3 (x = 2.9), the expression becomes: (2(2.9) + 1)/(2.9 - 3) = 5.8/-0.1 = -58
When x is slightly greater than 3 (x = 3.1), the expression becomes: (2(3.1) + 1)/(3.1 - 3) = 6.2/0.1 = 62
As x gets closer to 3, the expression approaches either positive or negative infinity.
Therefore, the limit as x approaches 3 of (2x + 1)/(x-3) does not exist.
For problem 9:
We substitute x values that approach 0:
When x is slightly less than 0 (x = -0.1), the expression becomes: 2/(-0.1 + 1) = 2/0.9 = 2.22...
When x is slightly greater than 0 (x = 0.1), the expression becomes: 2/(0.1 + 1) = 2/1.1 = 1.81...
As x gets closer to 0, the expression approaches 2.
Therefore, the limit as x approaches 0 of (2/(x+1))/(x-0) is equal to 2.