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Exercise

$\lim_{x\to infinity}\left(\csc\left(\frac{1}{x}\right)e^{\frac{1}{x}}-1\right)$

Step-by-step Solution

1

Applying rationalisation

$\lim_{x\to\infty }\left(\left(e^{\frac{1}{x}}\csc\left(\frac{1}{x}\right)-1\right)\frac{e^{\frac{1}{x}}\csc\left(\frac{1}{x}\right)+1}{e^{\frac{1}{x}}\csc\left(\frac{1}{x}\right)+1}\right)$
2

Multiply and simplify the expression within the limit

$\lim_{x\to\infty }\left(\frac{\left(e^{\frac{1}{x}}\csc\left(\frac{1}{x}\right)\right)^2-1}{e^{\frac{1}{x}}\csc\left(\frac{1}{x}\right)+1}\right)$
3

The power of a product is equal to the product of it's factors raised to the same power

$\lim_{x\to\infty }\left(\frac{e^{\frac{2}{x}}\csc\left(\frac{1}{x}\right)^2-1}{e^{\frac{1}{x}}\csc\left(\frac{1}{x}\right)+1}\right)$
4

The difference of the squares of two terms, divided by the sum of the same terms, is equal to the difference of the terms. In other words: $\displaystyle\frac{a^2-b^2}{a+b}=a-b$.

$\lim_{x\to\infty }\left(c-f\right)$
5

The limit of a constant is just the constant

$c-f$

Final answer to the exercise

$c-f$

Try other ways to solve this exercise

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  • Solve using limit properties
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  • Solve the limit using factorization
  • Solve the limit using rationalization
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
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