lim x approaches 64 6th root of only x and -2/ cube root of only x and -4
I read that as
lime (x^(1/6) - 2)/(x^(1/3) - 4) as x ----> 64
let x^(1/6) = k
then x^(1/3) = k^2 , and as x ---> 64, k ---> 2
so we have:
lim (k-2)/(k^2 - 4) , as k ---> 2
= lim (k-2)/( (k-2)(k+2)) , k ---> 2
= lim 1/(k+2) , as k ---> 2
= 1/4
solve it plz
To evaluate the limit, let's break it down into two separate parts:
1. The limit of the 6th root of x as x approaches 64.
The 6th root of x can be written as x^(1/6). So, we have:
lim (x → 64) (x^(1/6))
To evaluate this limit, we can simply substitute the value 64 into the expression:
(64)^(1/6) = 2
Therefore, the limit of the 6th root of x as x approaches 64 is equal to 2.
2. The limit of -2 divided by the cube root of x minus 4 as x approaches 64.
The expression can be written as:
lim (x → 64) (-2 / (x^(1/3) - 4))
Again, we can directly substitute the value 64 into the expression:
-2 / ((64)^(1/3) - 4)
Let's calculate the cube root of 64:
(64)^(1/3) = 4
Now we substitute this value into our expression:
-2 / (4 - 4) = -2 / 0
Division by zero is undefined, so the limit does not exist.
In summary, the limit of the 6th root of x as x approaches 64 is 2, and the limit of -2 divided by the cube root of x minus 4 as x approaches 64 does not exist.
To find the limit of a function as x approaches a certain value, we substitute the value into the function and see what the function approaches. Let's start with the given expressions:
The first expression is the 6th root of only x.
lim x→64 (6th root of only x)
To simplify, we can rewrite the 6th root as the exponent of 1/6:
lim x→64 (x^(1/6))
Now we substitute 64 into the expression:
(64)^(1/6)
64 to the power of 1/6 can be rewritten as (2^6)^(1/6) since 64 is equal to 2 raised to the power of 6:
(2^6)^(1/6) = 2^((6*1)/6) = 2^1 = 2
So, the first expression simplifies to 2.
Now let's move on to the second expression:
The second expression is -2 divided by the cube root of only x and -4.
lim x→64 (-2 / (cube root of only x and -4))
To simplify, we can rewrite the cube root as the exponent of 1/3:
lim x→64 (-2 / (x^(1/3) - 4))
Now we substitute 64 into the expression:
-2 / (64^(1/3) - 4)
64 to the power of 1/3 can be rewritten as (4^3)^(1/3) since 64 is equal to 4 raised to the power of 3:
(4^3)^(1/3) = 4^((3*1)/3) = 4^1 = 4
So, the expression simplifies to:
-2 / (4 - 4)
Since 4 - 4 equals 0, we have:
-2 / 0
Division by zero is undefined, so we cannot determine the value of this expression.
In summary, the first expression simplifies to 2, while the second expression is undefined.