Evaluat d 4lowing1.lim x/|x| x-->0
2.lim x->1 sqrt(x^2+2- sqrt3)/x-1
3.lim n->~ f(n)=(1+1/n)^sqrtn
4. limx->0 f(x)= (12^x-3^x-4^x+1)/xtanx
5. Lim x->3 (x^n-3^n)^n/(n-3)^n
Sure! Let's evaluate each of these limits step by step.
1. lim(x→0) x/|x|
To evaluate this limit, we need to consider the left-hand and right-hand limits separately. Let's start with the left-hand limit (approaching 0 from the negative side). For x < 0, we have |x| = -x, so the expression becomes lim(x→0-) x/(-x). Simplifying this, we get lim(x→0-) -1. Therefore, the left-hand limit is -1.
Now let's consider the right-hand limit (approaching 0 from the positive side). For x > 0, we have |x| = x, so the expression becomes lim(x→0+) x/x. Simplifying this, we get lim(x→0+) 1. Therefore, the right-hand limit is 1.
Since the left and right-hand limits do not match, the overall limit does not exist for this function.
2. lim(x→1) sqrt(x^2+2 - sqrt(3))/(x-1)
To evaluate this limit, we can start by simplifying the numerator. We have sqrt(x^2+2 - sqrt(3)) = sqrt(x^2 - sqrt(3) + 2). Factoring out the square root of 3, we get sqrt(3)(sqrt(3)/3) = sqrt(3)/sqrt(3) = 1. Therefore, the numerator simplifies to 1.
Now we have lim(x→1) 1/(x-1). Plugging in x = 1, we get 1/(1-1) = 1/0. Since division by zero is undefined, the limit does not exist.
3. lim(n→∞) f(n) = (1+1/n)^(sqrt(n))
To evaluate this limit, we need to analyze the behavior as n approaches infinity. Notice that (1+1/n)^(sqrt(n)) can be rewritten as ((1+1/n)^(n))^(1/√n).
Now we recognize the term (1+1/n)^n as the definition of the mathematical constant e. Therefore, as n approaches infinity, (1+1/n)^n approaches e. Additionally, we have (1/√n) approaching 0 as n approaches infinity.
Putting it all together, we get lim(n→∞) f(n) = e^0 = 1.
4. lim(x→0) f(x) = (12^x - 3^x - 4^x + 1)/(x * tan(x))
To evaluate this limit, we can start by factoring out the common factor in the numerator, which is 1: f(x) = (1/1)(12^x - 3^x - 4^x + 1)/(x * tan(x)).
Now we can evaluate the limit term by term:
lim(x→0) 1 = 1
lim(x→0) 12^x - 3^x - 4^x + 1 = 12^0 - 3^0 - 4^0 + 1 = 1 - 1 - 1 + 1 = 0
lim(x→0) x = 0
lim(x→0) tan(x) = tan(0) = 0
Putting it all together, we get lim(x→0) f(x) = (0)/(0) = undefined or indeterminate.
5. lim(x→3) (x^n - 3^n)^n/(n-3)^n
To evaluate this limit, we can start by factoring out the common factor in the numerator, which is x^n - 3^n: f(x) = ((x^n - 3^n)^n)/(n - 3)^n.
Now we can evaluate the limit term by term:
lim(x→3) x^n - 3^n = 3^n - 3^n = 0
lim(x→3) n - 3 = n - 3
Taking the limit as n approaches infinity gives us n - 3 = ∞
Putting it all together, we get lim(x→3) f(x) = ((0)^∞)/(∞) = 0/∞ = 0.
I hope this helps! Let me know if you have any further questions.