Understanding the Limit of sin(ax)/sin(bx) for Real Numbers a and b

When evaluating the limit of the function (frac{sin(ax)}{sin(bx)}) for real numbers (a) and (b), different scenarios arise depending on the values of (a) and (b). This article delves into the various cases and explains the mathematical reasoning behind each scenario.

1. When (b0)

If (b0), the function (frac{sin(ax)}{sin(bx)}) is never defined, and therefore, there is no limit. This is because the denominator becomes zero, and any non-zero numerator divided by zero is undefined.

2. When (b) is an Integer and (a) is Not an Integer

If (b) is an integer but (a) is not, the limit of (frac{sin(ax)}{sin(bx)}) as (x) approaches (pi) does not exist. This is due to the fact that as (x) approaches (pi) from one side, (sin(bx)) approaches zero, making the overall expression grow larger than any real number. On the other side, (sin(bx)) also approaches zero but in a different direction, making the expression smaller than any real number. Thus, the function oscillates infinitely and the limit does not exist.

3. When Neither (a) nor (b) are Integers

If neither (a) nor (b) are integers, the limit of (frac{sin(ax)}{sin(bx)}) as (x) approaches zero can be evaluated directly using the formula (frac{sin(ax)}{sin(bx)} frac{a sin(ax)}{b sin(bx)}) and simplifying the expression.

3.1 Simplification Using Small Angle Approximation

For values close to zero, using the small angle approximation (sin(z) approx z), the expression can be simplified as follows:

Let (u x - pi) Using the angle addition formula for sine, (sin(a(x - pi)) -1^a sin(ax)). Similarly, (sin(b(x - pi)) -1^b sin(bx)).

Therefore, the limit becomes:

(lim_{x to 0} frac{sin(ax)}{sin(bx)} -1^{ab} frac{a}{b})

4. When Both (a) and (b) are Integers

If both (a) and (b) are integers, we use the small angle approximation technique to simplify the expression:

For values close to zero, we have (sin(cx) approx cx). Let (u x - pi) and use the angle addition formula for sine:

(frac{sin(ax)}{sin(bx)} frac{sin(a(u pi))}{sin(b(u pi))} frac{sin(au api)}{sin(bu bpi)} -1^{ab} frac{sin(au)}{sin(bu)})

Since (sin(au) au O(x^3)) and (sin(bu) bu O(x^3)), the limit as (x to 0) is:

(lim_{x to 0} frac{sin(ax)}{sin(bx)} -1^{ab} frac{a}{b})

5. Direct Calculation for Non-Integer a and b

If (a) and (b) are not both integers, you can directly substitute the values and evaluate the limit without further manipulation. This is because the expression (frac{sin(ax)}{sin(bx)}) simplifies to a constant value.

6. L'Hospital's Rule for Integer a and b

If (a) and (b) are both integers, using L'H?pital's rule is particularly useful to evaluate the limit as it simplifies the differentiation process:

(lim_{x to 0} frac{sin(ax)}{sin(bx)} lim_{x to 0} frac{a cos(ax)}{b cos(bx)} frac{a}{b})

This is due to the fact that (cos(ax)) and (cos(bx)) are bounded and do not approach zero as (x) approaches zero.

7. Conclusion

Understanding the limit of (frac{sin(ax)}{sin(bx)}) requires recognizing the different scenarios based on the values of (a) and (b). Whether you use small angle approximations, direct substitution, or L'H?pital's rule, the key is to simplify the expression and evaluate the limit accurately.