Find
lim θ→0+ A(θ)/B(θ)
To find the limit θ→0+ A(θ)/B(θ), we can follow these steps:
Step 1: Determine the functions A(θ) and B(θ) involved in the limit.
Step 2: Substitute the value 0 into both functions A(θ) and B(θ) to evaluate them at the limit θ→0+.
Step 3: Simplify the expression A(θ)/B(θ) as much as possible.
Step 4: Evaluate the resulting expression at the limit θ→0+.
Let's go through an example to illustrate these steps:
Example: Find lim θ→0+ (sin(θ))/θ
Step 1: The functions involved in the limit are A(θ) = sin(θ) and B(θ) = θ.
Step 2: Substitute θ = 0 into A(θ) and B(θ):
A(0) = sin(0) = 0
B(0) = 0
Step 3: From the previous step, we have A(0) = 0 and B(0) = 0. However, we encounter an indeterminate form of 0/0.
Step 4: To resolve this, we can use L'Hôpital's Rule, which states that if we have an indeterminate form of 0/0, we can take the derivative of the numerator and the derivative of the denominator, and then evaluate the limit again. Applying this rule, we have:
lim θ→0+ (sin(θ))/θ = lim θ→0+ (cos(θ))/1
We obtain this result by taking the derivative of the numerator (sin(θ)) and the derivative of the denominator (θ), which gives us cos(θ)/1 = cos(θ).
Step 4 (continued): Now, we can evaluate the expression at the limit θ→0+, which gives us:
lim θ→0+ (cos(θ))/1 = cos(0) = 1
Therefore, the limit of (sin(θ))/θ as θ approaches 0 from the positive side is 1.
Note: L'Hôpital's Rule can be applied in certain cases when dealing with indeterminate forms such as 0/0 or ∞/∞. However, it should be used judiciously, and not all indeterminate forms can be resolved by using it.