Evaluate the limit using L'Hospital's rule
$\lim _{x \rightarrow 0} \frac{\sin (9 x)}{\tan (15 x)}$
Answer (2)
Recognize the limit is in 0 0 indeterminate form.
Apply L'Hospital's rule by taking the derivative of the numerator and denominator: d x d ( sin ( 9 x )) = 9 cos ( 9 x ) and d x d ( tan ( 15 x )) = 15 sec 2 ( 15 x ) .
Evaluate the new limit: lim x → 0 15 s e c 2 ( 15 x ) 9 c o s ( 9 x ) = 15 9 .
Simplify the result to get the final answer: 5 3 .
Explanation
Problem Setup We are asked to evaluate the limit lim x → 0 t a n ( 15 x ) s i n ( 9 x ) using L'Hospital's rule.
Indeterminate Form First, we check if the limit is in an indeterminate form. As x approaches 0, sin ( 9 x ) approaches sin ( 0 ) = 0 and tan ( 15 x ) approaches tan ( 0 ) = 0 . Thus, the limit is of the indeterminate form 0 0 .
L'Hospital's Rule Since we have an indeterminate form, we can apply L'Hospital's rule, which states that if lim x → c g ( x ) f ( x ) is of the form 0 0 or ∞ ∞ , then lim x → c g ( x ) f ( x ) = lim x → c g ′ ( x ) f ′ ( x ) , provided the limit exists.
Derivatives We need to find the derivatives of the numerator and the denominator. The derivative of sin ( 9 x ) with respect to x is 9 cos ( 9 x ) . The derivative of tan ( 15 x ) with respect to x is 15 sec 2 ( 15 x ) .
New Limit Now we can rewrite the limit using the derivatives: lim x → 0 15 s e c 2 ( 15 x ) 9 c o s ( 9 x ) .
Evaluating the Limit Next, we evaluate the limit by substituting x = 0 into the expression: 15 s e c 2 ( 15 ⋅ 0 ) 9 c o s ( 9 ⋅ 0 ) = 15 s e c 2 ( 0 ) 9 c o s ( 0 ) = 15 ⋅ 1 2 9 ⋅ 1 = 15 9 .
Simplifying Finally, we simplify the fraction 15 9 to 5 3 .
Final Answer Therefore, the limit is 5 3 .
Examples
L'Hopital's rule is often used in physics and engineering to analyze systems where quantities approach indeterminate forms. For example, in circuit analysis, you might encounter a situation where both voltage and current approach zero, and you need to find the limit of their ratio to determine the resistance. Similarly, in fluid dynamics, when analyzing the flow near a singularity, L'Hopital's rule can help determine the behavior of the fluid by evaluating indeterminate forms involving velocity and pressure. This rule provides a powerful tool for understanding the behavior of complex systems at critical points.
The limit lim x → 0 t a n ( 15 x ) s i n ( 9 x ) can be evaluated using L'Hospital's rule. This gives the result of 5 3 . Therefore, the final answer is 5 3 .
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