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Exercise

$\lim_{x\to\pi}\left(\frac{cos\left(x\right)+sin\left(2x\right)+1}{x^2-\pi^2}\right)$

Step-by-step Solution

1

Evaluate the limit $\lim_{x\to\pi }\left(\frac{\cos\left(x\right)+\sin\left(2x\right)+1}{x^2- \pi ^2}\right)$ by replacing all occurrences of $x$ by $\pi $

$\frac{\cos\left(\pi \right)+\sin\left(2\pi \right)+1}{\pi ^2- \pi ^2}$
2

The cosine of $\pi $ equals $-1$

$\frac{-1+\sin\left(2\pi \right)+1}{\pi ^2- \pi ^2}$
3

Subtract the values $1$ and $-1$

$\frac{\sin\left(2\pi \right)}{\pi ^2- \pi ^2}$

Final answer to the exercise

$\frac{\sin\left(2\pi \right)}{\pi ^2- \pi ^2}$

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
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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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