Evaluate limit of 1/(x⁴+6x+2) as x approaches 0
To evaluate the limit of 1/(x⁴+6x+2) as x approaches 0, we substitute 0 for x and simplify:
1/(0⁴+6(0)+2) = 1/2
Therefore, the limit of 1/(x⁴+6x+2) as x approaches 0 is 1/2.
evaluate the limit f(x) ={x+3, x≥0 and x-3, x<0} as x approaches 0
To evaluate the limit of f(x) as x approaches 0, we need to determine the left and right limits.
From the definition of f(x), we have:
- For x ≥ 0, f(x) = x + 3
- For x < 0, f(x) = x - 3
Therefore, the left limit of f(x) as x approaches 0 (i.e. the limit as x approaches 0 from the left-hand side) is:
lim x→0- f(x) = lim x→0- (x - 3) = -3
This is because for x < 0 and approaching 0, f(x) approaches (0-3) = -3.
Similarly, the right limit of f(x) as x approaches 0 (i.e. the limit as x approaches 0 from the right-hand side) is:
lim x→0+ f(x) = lim x→0+ (x + 3) = 3
This is because for x ≥ 0 and approaching 0, f(x) approaches (0+3) = 3.
Since the left and right limits are different, the limit of f(x) as x approaches 0 does not exist.
To evaluate the limit of 1/(x⁴+6x+2) as x approaches 0, we need to substitute x = 0 into the function and see what value it approaches.
So, let's plug in x = 0 into the function:
lim[x→0] (1/(x⁴+6x+2))
Now, substitute x = 0:
lim[x→0] (1/(0⁴+6(0)+2))
Simplifying further:
lim[x→0] (1/(0+0+2))
lim[x→0] (1/2)
Therefore, the limit of 1/(x⁴+6x+2) as x approaches 0 is 1/2.