4. lim (tanx)=
x->pi/3
-(sqrt3)
1
(sqrt3)
***-1
The limit does not exist.
5. lim |x|=
x->-2
-2
***2
0
-1
The limit does not exist.
6. lim [[x]]=
x->9/2
(Remember that [[x]] represents the greatest integer function of x.)
4
5
***4.5
-4
The limit does not exist.
7. lim [[x]]=
x->9
9
8
7
10
***The limit does not exist.
#4 surely you know that tan π/3 = √3
#5 ok
#6[[x]] represents the greatest integer function
How could you choose 4.5?
#7 ok
To find the limit of a function as x approaches a specific value, you need to evaluate the function as x gets arbitrarily close to that value.
For the first question, finding the limit of tanx as x approaches π/3, you can write the limit definition as:
lim (tanx) = lim (sinx/cosx)
x->π/3 x->π/3
Using the trigonometric identity, tanx = sinx/cosx, you can rewrite the expression as:
lim (sinx) 1
x->π/3 x->π/3 lim ( ---- )
x->π/3 cosx
Evaluating the individual limits:
lim (sinx) = sin(π/3) = √3/2
x->π/3
lim (cosx) = cos(π/3) = 1/2
x->π/3
Therefore, the limit is (√3/2) / (1/2) = √3.
For the second question, finding the limit of |x| as x approaches -2, you can use the definition of the absolute value function:
lim |x| = |lim x|
x->-2 x->-2
Since the limit of x as x approaches -2 is -2, the limit of |x| is also -2.
For the third question, finding the limit of [[x]] (greatest integer function) as x approaches 9/2, you need to consider that the greatest integer function rounds a given value down to the nearest integer. Since 9/2 is between 4 and 5, the limit of [[x]] as x approaches 9/2 is 4.
For the fourth question, finding the limit of [[x]] as x approaches 9, the greatest integer function will always round down to the nearest integer. Therefore, the limit does not exist since it approaches different values from the left and right sides of 9.
Remember that identifying whether a limit exists or not depends on the behavior of the function as x approaches the specific value, and can also be affected by any discontinuities or jumps in the function.