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Exercise

$\lim_{x\to\infty}\left(-\log\left(x+2\right)+\log\left(x-2\right)\right)$

Step-by-step Solution

1

Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\lim_{x\to\infty }\left(\frac{-\ln\left(x+2\right)}{\ln\left(10\right)}+\log \left(x-2\right)\right)$
2

Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\lim_{x\to\infty }\left(\frac{-\ln\left(x+2\right)}{\ln\left(10\right)}+\frac{\ln\left(x-2\right)}{\ln\left(10\right)}\right)$
3

Combine fractions with common denominator $\ln\left(10\right)$

$\lim_{x\to\infty }\left(\frac{-\ln\left(x+2\right)+\ln\left(x-2\right)}{\ln\left(10\right)}\right)$
4

Evaluate the limit $\lim_{x\to\infty }\left(\frac{-\ln\left(x+2\right)+\ln\left(x-2\right)}{\ln\left(10\right)}\right)$ by replacing all occurrences of $x$ by $\infty $

indeterminate

Final answer to the exercise

indeterminate

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
  • FOIL Method
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Main Topic: Operations with Infinity

Indications for how we should operate with infinity. Very useful to solve limits.

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