High School

We appreciate your visit to Evaluate the following limits a lim x 0 frac x 9 x 9 x 1 b lim x infinity x 7 tan 9 x 7. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Evaluate the following limits. (a) lim _{x → 0} \frac{x 9^{x}}{9^{x}-1} (b) lim _{x → [infinity]} x^{7} tan (9 / x^{7}) (c) lim _{x → [infinity]}\le

Answer :

(a) [tex]The limit lim_{x → ∞} x^{7} tan(9 / x^{7}) does not exist. (b) limit lim_{x → 0} \frac{x 9^{x}}{9^{x}-1} is approximately 0.454545455.[/tex]

(a) To evaluate the limit lim_{x → 0} \frac{x 9^{x}}{9^{x}-1}, we can use L'Hôpital's rule. This rule states that if we have an indeterminate form of the type 0/0 or ∞/∞, we can differentiate the numerator and the denominator separately until we obtain a non-indeterminate form.

Let's apply this rule to our limit. Taking the derivative of the numerator and the denominator separately, we have:

[tex]lim_{x → 0} \frac{9^{x} + x\cdot 9^{x}\ln(9)}{9^{x}\ln(9)}Now, plugging in x = 0 into the expression above, we get:lim_{x → 0} \frac{9^{0} + 0\cdot 9^{0}\ln(9)}{9^{0}\ln(9)}[/tex]

Simplifying this expression, we have:

[tex]lim_{x → 0} \frac{1 + 0}{\ln(9)}Since 1 divided by any nonzero number is still 1, and the denominator is a constant, the limit simplifies to:lim_{x → 0} \frac{1}{\ln(9)}Calculating the value of ln(9) ≈ 2.197224577, we can evaluate the limit as:lim_{x → 0} \frac{1}{\ln(9)} ≈ \frac{1}{2.197224577} ≈ 0.454545455Therefore, the limit lim_{x → 0} \frac{x 9^{x}}{9^{x}-1} is approximately 0.454545455.[/tex]

(b) To evaluate the limit lim_{x → ∞} x^{7} tan(9 / x^{7}), we need to analyze the behavior of the function as x approaches infinity.

As x gets larger and larger, the term x^{7} dominates the expression. The tangent function, tan(9 / x^{7}), oscillates between -1 and 1 as the argument approaches 0. Therefore, multiplying x^{7} by a value that oscillates between -1 and 1 will result in the limit being undefined as x approaches infinity.

In other words, the limit lim_{x → ∞} x^{7} tan(9 / x^{7}) does not exist.

Learn more about limit on

https://brainly.com/question/30679261

#SPJ11

Thanks for taking the time to read Evaluate the following limits a lim x 0 frac x 9 x 9 x 1 b lim x infinity x 7 tan 9 x 7. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada