Find the limits
Lim(sin kt/t) t approaches 0 and (K is constant)
How do you start this and go about it stepwise?
Does knowing (K constant) help me to solve this?
You can do this in one step.
As t -> 0, sinkt -> kt, so
Lim (sinkt)/t = kt/t = k
t->0
You could also use l'Hopital's rule and get the same answer.
To find the limit of the function Lim(sin(kt)/t) as t approaches 0, you can use the method of L'Hôpital's Rule. The constant, k, does not directly affect the solution.
Here are the step-by-step instructions:
Step 1: Start by substituting 0 into the function expression. This gives us sin(0k)/0, which is 0/0 form, an indeterminate form.
Step 2: Apply L'Hôpital's Rule, which states that if you have an indeterminate form of 0/0 or ∞/∞, you can evaluate the limit by taking the derivative of the numerator and denominator and then re-evaluating the limit.
Step 3: Take the derivative of the numerator and denominator separately. The derivative of sin(kt) with respect to t is k*cos(kt), and the derivative of t is 1.
Step 4: Now re-evaluate the limit after taking the derivative. We now have Lim((k*cos(kt))/1) as t approaches 0.
Step 5: Substitute 0 into the derivative expression. Since cosine of 0 is equal to 1, the expression becomes k*1/1, which simplifies to k.
Therefore, the limit of Lim(sin(kt)/t) as t approaches 0 is k.
To find the limit of the given function, Lim(sin(kt)/t) as t approaches 0, you can start by applying the sin(x)/x limit. This limit states that Lim(sin(x)/x) as x approaches 0 is equal to 1.
Now, let's apply this to our problem step by step:
1. Substitute kt for x: Lim(sin(kt)/t) as t approaches 0 becomes Lim(sin(kt)/(kt)) as t approaches 0.
2. Use the sin(x)/x limit: Now, let's apply the sin(x)/x limit to the expression: Lim(sin(kt)/(kt)) as t approaches 0 = 1 * Lim(1) as t approaches 0.
3. Simplify the expression: Lim(1) as t approaches 0 is equal to 1.
Therefore, the limit of sin(kt)/t as t approaches 0 is 1.
Having k as a constant does not affect the limit in this case. It only specifies that k is not a variable, but a fixed constant throughout the problem. So, knowing that k is constant does not change the way you solve the limit.