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Exercise

$\lim_{x\to 5}\left(\frac{x^2-25}{x-5}\right)$

Step-by-step Solution

1

If we directly evaluate the limit $\lim_{x\to5}\left(\frac{x^2-25}{x-5}\right)$ as $x$ tends to $5$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
2

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

$\lim_{x\to 5}\left(\frac{\frac{d}{dx}\left(x^2-25\right)}{\frac{d}{dx}\left(x-5\right)}\right)$
3

After deriving both the numerator and denominator, and simplifying, the limit results in

$\lim_{x\to5}\left(2x\right)$
4

Evaluate the limit $\lim_{x\to5}\left(2x\right)$ by replacing all occurrences of $x$ by $5$

$2\cdot 5$
5

Multiply $2$ times $5$

$10$

Final answer to the exercise

$10$

Try other ways to solve this exercise

  • 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
  • FOIL Method
  • Integrate by substitution
  • Load more...
Can't find a method? Tell us so we can add it.

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Main Topic: Limits by L'Hôpital's rule

In mathematics, and more specifically in calculus, L'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms.

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