Find the limit. (Hint: Let [tex]$x=\frac{1}{t}$[/tex] and find the limit as [tex]$t \rightarrow 0^{+}$[/tex]. If an answer does not exist, enter DNE.)
[tex]$\lim _{x \rightarrow \infty} 8 x \tan \left(\frac{9}{x}\right)$[/tex]
Answer (2)
Substitute x = t 1 , transforming the limit to lim t → 0 + t 8 t a n ( 9 t ) .
Recognize the indeterminate form 0 0 and apply L'Hopital's rule.
Differentiate the numerator and denominator to get lim t → 0 + 1 72 s e c 2 ( 9 t ) .
Evaluate the limit to find the answer: 72 .
Explanation
Problem Setup We are asked to find the limit of the function 8 x tan ( x 9 ) as x approaches infinity. We are given a hint to use the substitution x = t 1 and find the limit as t approaches 0 from the positive side.
Substitution Let's substitute x = t 1 into the expression. Then, as x → ∞ , t → 0 + . The expression becomes:
8 x tan ( x 9 ) = 8 ( t 1 ) tan ( 9 t ) = t 8 t a n ( 9 t )
Rewriting the Limit Now we need to find the limit as t → 0 + :
lim x → ∞ 8 x tan ( x 9 ) = lim t → 0 + t 8 t a n ( 9 t )
Applying L'Hopital's Rule We can see that as t → 0 + , the expression becomes of the form 0 0 , which is an indeterminate form. Therefore, we can use L'Hopital's rule, which 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.
Finding Derivatives Let's find the derivatives of the numerator and the denominator with respect to t :
Numerator: f ( t ) = 8 tan ( 9 t ) . The derivative is f ′ ( t ) = 8 ⋅ 9 sec 2 ( 9 t ) = 72 sec 2 ( 9 t ) .
Denominator: g ( t ) = t . The derivative is g ′ ( t ) = 1 .
Applying the Derivatives Now we can rewrite the limit using L'Hopital's rule:
lim t → 0 + t 8 t a n ( 9 t ) = lim t → 0 + 1 72 s e c 2 ( 9 t )
Evaluating the Limit Now we evaluate the limit as t → 0 + :
lim t → 0 + 72 sec 2 ( 9 t ) = 72 sec 2 ( 0 ) = 72 ⋅ 1 = 72
Final Answer Therefore, the limit of the given expression as x approaches infinity is 72.
lim x → ∞ 8 x tan ( x 9 ) = 72
Examples
Imagine you are designing a telescope and need to align its mirrors precisely. The tangent function appears in calculations involving angles of light reflection. Determining limits involving the tangent function, similar to this problem, helps ensure that the telescope focuses correctly on distant stars. This ensures minimal distortion and maximum clarity in astronomical observations.
The limit of the expression as x approaches infinity is found to be 72 using the substitution x = 1/t and applying L'Hopital's rule. After transforming the limit and differentiating, we arrive at the final answer. Thus, lim x → ∞ 8 x tan ( x 9 ) = 72.
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