Question

Calculate lim sin(5x)/sin(2x)

x→0 . Do not use L'Hôpital's rule.
a) 2.5
b) 5
c) 0
d)1

Answer 1

The limit of sin(5x)/sin(2x) as x approaches 0 is found by rewriting it in terms of the limit of sin(θ)/θ as θ approaches 0, yielding a final result of 2.5. The correct answer is b) 2.5.

Step-by-step explanation:

The student is asking to calculate the limit of the function sin(5x)/sin(2x) as x approaches 0. This type of problem is common in calculus and involves understanding the behavior of functions as they approach certain points.

To solve this without using L'Hôpital's rule, we can use a well-known trigonometric limit: ℓ sin(θ)/θ = 1 as θ approaches 0. By substituting 5x for θ in the numerator and 2x for θ in the denominator, and since x is approaching 0, both 5x and 2x are approaching 0 as well. Therefore, their respective sine functions will behave similarly to their arguments.

We can rewrite the original limit as:

(5/2) × (sin(5x)/(5x)) × (2x/sin(2x))

And as x approaches 0:

lim (sin(5x)/(5x)) = 1

lim (2x/sin(2x)) = 1

So the limit of the original function as x approaches 0 is just:

lim sin(5x)/sin(2x) = (5/2) × 1 × 1 = 5/2 = 2.5

Therefore, the correct answer is b) 2.5.