Finding the Limit of an Arctan Equation

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SUMMARY

The limit of the equation arctan(3/(x-2)) as x approaches 2 from the left is evaluated by recognizing that as x approaches 2, the expression 3/(x-2) approaches negative infinity. Consequently, the limit is arctan(-∞), which equals -π/2. The discussion highlights the importance of understanding one-to-one functions and the application of limit properties in calculus.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with the arctangent function
  • Knowledge of one-to-one functions
  • Basic algebraic manipulation of functions
NEXT STEPS
  • Study the properties of the arctangent function and its limits
  • Learn about one-to-one functions and their implications in calculus
  • Explore advanced limit techniques, including L'Hôpital's Rule
  • Practice solving limits involving rational functions
USEFUL FOR

Students studying calculus, particularly those focusing on limits and trigonometric functions, as well as educators seeking to enhance their teaching methods in these areas.

greenglasses
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The Question
Find the limit as x approaches 2 from the left for the equation
arctan(3/(x-2))
Work Done
I'm not really sure even where to start. My teacher didn't go over this type of question. I don't see any way to mess with the equation to make it so we can plug in 2, and I don't see how rewriting the equation in the form tan y = 3/(x-2) could help either. I'm not supposed to take the derivative, am I?
I would appreciate any pushes in the right direction.
 
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I would begin with the fact that given a one-to-one function $f$, we may state:

$$\lim_{x\to a}\left(f\left(u(x) \right) \right)=f\left(\lim_{x\to a}\left(u(x) \right) \right)$$
 
Thanks, I managed to figure it out though.
 

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