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Let's solve the given problems by evaluating the limits as specified.

### 17. \(\lim_{{x \to 2}} \frac{1}{{x^2 - 4}}\)
The function is \(\frac{1}{{x^2 - 4}}\), which can be rewritten as \(\frac{1}{{(x-2)(x+2)}}\).

**As \(x \to 2^+\)**:
- As \(x\) approaches 2 from the right, \((x-2)\) is positive but close to zero, and \((x+2)\) is positive. Therefore, \(\frac{1}{(x-2)(x+2)}\) approaches \(+\infty\).

**As \(x \to 2^-\)**:
- As \(x\) approaches 2 from the left, \((x-2)\) is negative but close to zero, and \((x+2)\) is positive. Therefore, \(\frac{1}{(x-2)(x+2)}\) approaches \(-\infty\).

**As \(x \to -2^+\)**:
- As \(x\) approaches \(-2\) from the right, \((x+2)\) is positive but close to zero, and \((x-2)\) is negative. Therefore, \(\frac{1}{(x-2)(x+2)}\) approaches \(-\infty\).

**As \(x \to -2^-\)**:
- As \(x\) approaches \(-2\) from the left, \((x+2)\) is negative and close to zero, and \((x-2)\) is negative. Therefore, \(\

Find the limits in the following exercises: 
17. lim (1 / (x^2 - 4)) as x approaches 2 and -2 from both left and right. 
18. lim (x / (x^2 - 1)) as x approaches 1 from both left and right. 
19. lim ((x^2 / 2) - (1 / x)) as x approaches 0 and other values. 
20. lim ((x^2 / 2x + 4)) as x approaches -2 and 1 from both sides.

Detailed solutions for limit calculations involving rational and polynomial functions. The problems cover approaching values from both sides, using indeterminate forms, and analyzing asymptotic behavior. Perfect for Grade 11-12 students preparing for calculus.

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