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Exercise

$\lim_{x\to\frac{2}{\pi}}\left(\frac{\tan x}{\tan5x}\right)$

Step-by-step Solution

1

Evaluate the limit $\lim_{x\to{\frac{2}{\pi }}}\left(\frac{\tan\left(x\right)}{\tan\left(5x\right)}\right)$ by replacing all occurrences of $x$ by $\frac{2}{\pi }$

$\frac{\tan\left(\frac{2}{\pi }\right)}{\tan\left(5\cdot \left(\frac{2}{\pi }\right)\right)}$
2

Multiply the fraction and term in $5\cdot \left(\frac{2}{\pi }\right)$

$\frac{\tan\left(\frac{2}{\pi }\right)}{\tan\left(\frac{5\cdot 2}{\pi }\right)}$
3

Multiply $5$ times $2$

$\frac{\tan\left(\frac{2}{\pi }\right)}{\tan\left(\frac{10}{\pi }\right)}$

Final answer to the exercise

$\frac{\tan\left(\frac{2}{\pi }\right)}{\tan\left(\frac{10}{\pi }\right)}$

Try other ways to solve this exercise

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  • Solve using L'Hôpital's rule
  • Solve without using l'Hôpital
  • Solve using limit properties
  • Solve using direct substitution
  • Solve the limit using factorization
  • Solve the limit using rationalization
  • Integrate by partial fractions
  • Product of Binomials with Common Term
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Main Topic: Limits by Direct Substitution

Find limits of functions at a specific point by directly plugging the value into the function.

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