$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{\cos x}$
Answer (2)
The limit lim x → 2 π c o s x 1 − s i n x evaluates to 0, after confirming it is an indeterminate form and applying L'Hopital's Rule. By differentiating the numerator and denominator, we find the limit ultimately leads to 0 .
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Recognize the limit is in 0 0 indeterminate form.
Apply L'Hopital's rule by differentiating the numerator and the denominator.
Find the derivatives: ( 1 − sin x ) ′ = − cos x and ( cos x ) ′ = − sin x .
Evaluate the limit of the derivatives: lim x → 2 π − s i n x − c o s x = 0 .
Explanation
Problem Setup We are asked to find the limit of the function c o s x 1 − s i n x as x approaches 2 π .
Checking the Form First, let's check the form of the limit by plugging in x = 2 π into the function. We have sin ( 2 π ) = 1 and cos ( 2 π ) = 0 . Thus, the function becomes 0 1 − 1 = 0 0 , which is an indeterminate form. This means we can apply L'Hopital's rule.
Applying L'Hopital's Rule L'Hopital's rule states that if lim x → a g ( x ) f ( x ) is of the form 0 0 or ∞ ∞ , then lim x → a g ( x ) f ( x ) = lim x → a g ′ ( x ) f ′ ( x ) , provided the limit exists. In our case, f ( x ) = 1 − sin x and g ( x ) = cos x .
Finding Derivatives Now, we need to find the derivatives of f ( x ) and g ( x ) . The derivative of f ( x ) = 1 − sin x is f ′ ( x ) = − cos x . The derivative of g ( x ) = cos x is g ′ ( x ) = − sin x .
Applying the Derivatives Applying L'Hopital's rule, we have lim x → 2 π c o s x 1 − s i n x = lim x → 2 π − s i n x − c o s x = lim x → 2 π s i n x c o s x .
Evaluating the Limit Now, we evaluate the limit by substituting x = 2 π into the expression s i n x c o s x . We have s i n ( 2 π ) c o s ( 2 π ) = 1 0 = 0 .
Final Answer Therefore, the limit is 0.
Final Answer 0
Examples
Imagine you're designing a ramp where the height and length are described by trigonometric functions as an angle changes. Finding the limit as the angle approaches a critical value, like 2 π , helps ensure a smooth transition and avoids abrupt changes in the ramp's slope. This is crucial for safety and functionality, ensuring the ramp is usable and doesn't cause any unexpected issues.
