Evaluate the limit using L'Hopital's rule [tex] \lim _{x \rightarrow \infty} \frac{15 x^2}{e^{4 x}} [/tex]
Answer (1)
Recognize the limit is in the indeterminate form ∞ ∞ .
Apply L'Hopital's rule by differentiating the numerator and denominator, resulting in lim x → ∞ 4 e 4 x 30 x .
Apply L'Hopital's rule again, obtaining lim x → ∞ 16 e 4 x 30 .
Evaluate the limit as x approaches infinity, yielding 0 .
Explanation
Checking the Indeterminate Form We are asked to evaluate the limit lim x → ∞ e 4 x 15 x 2 using L'Hopital's rule. L'Hopital's rule is applicable when we have an indeterminate form of type 0 0 or ∞ ∞ . Let's check the form of the limit as x approaches infinity.
Confirming Indeterminate Form As x → ∞ , 15 x 2 → ∞ and e 4 x → ∞ . Thus, the limit is of the indeterminate form ∞ ∞ , and we can apply L'Hopital's rule.
First Application of L'Hopital's Rule Applying L'Hopital's rule once, we differentiate the numerator and the denominator with respect to x :
d x d ( 15 x 2 ) = 30 x d x d ( e 4 x ) = 4 e 4 x
So the limit becomes lim x → ∞ 4 e 4 x 30 x .
Checking the Indeterminate Form Again As x → ∞ , 30 x → ∞ and 4 e 4 x → ∞ . Thus, the limit is still of the indeterminate form ∞ ∞ , so we can apply L'Hopital's rule again.
Second Application of L'Hopital's Rule Applying L'Hopital's rule a second time, we differentiate the numerator and the denominator with respect to x :
d x d ( 30 x ) = 30 d x d ( 4 e 4 x ) = 16 e 4 x
So the limit becomes lim x → ∞ 16 e 4 x 30 .
Evaluating the Limit Now, as x → ∞ , e 4 x → ∞ , so 16 e 4 x 30 → 0 . Therefore, the limit is 0.
Final Answer Thus, lim x → ∞ e 4 x 15 x 2 = 0 .
Examples
L'Hopital's rule is not just a theoretical concept; it has practical applications in various fields. For instance, in electrical engineering, when analyzing the behavior of circuits with inductors and capacitors as time approaches infinity, you might encounter indeterminate forms. Applying L'Hopital's rule helps determine the circuit's steady-state behavior, ensuring designs meet performance requirements. Similarly, in chemical kinetics, when studying reaction rates that approach equilibrium, L'Hopital's rule can be used to evaluate complex rate expressions, providing insights into reaction mechanisms and optimizing industrial processes. These examples demonstrate how a seemingly abstract mathematical tool can be essential for solving real-world problems.
