FIND THE LIMIT
As x approaches 0:
tan 5 x/sin 25 x
use lptals rule
the derivative of tan(5x)=sec(5x)^2
the derivative of sin(25x)=cos(25x)*5
the lim of tan(5x)/(sin(25x))=the lim of
sec(5x)^2/(cos(25x)*5)
direct substution will work for this limit so I will evaluate at 0
the answer is 1/5
To find the limit of the given expression as x approaches 0, we can use trigonometric identities and basic limit properties.
The given expression is (tan 5x) / (sin 25x).
First, let's simplify the expression using trigonometric identities:
1. tan 5x = sin 5x / cos 5x
2. sin 25x = sin (5x * 5) = sin 5x * cos 5x + cos 5x * sin 5x
Now, substitute the simplified forms back into the original expression:
(sin 5x / cos 5x) / (sin 5x * cos 5x + cos 5x * sin 5x)
Next, simplify further by canceling out common factors:
1 / (cos 5x + sin 5x)
Now, let's find the limit of this expression as x approaches 0. We can substitute 0 into the expression and evaluate it:
1 / (cos 0 + sin 0) = 1 / (1 + 0) = 1 / 1 = 1
Therefore, as x approaches 0, the limit of (tan 5x) / (sin 25x) is equal to 1.