To determine the limit of any function as it approaches a particular value, follow these steps:

1. Direct Substitution:

• If the function is continuous at the point you're approaching (say, $x \to a$), substitute $x = a$ directly into the function $f(x)$.
• If you obtain a finite result, that's your limit.

Example: For $f(x) = x^2$ as $x \to 2$: $\lim_{{x \to 2}} x^2 = 2^2 = 4$

2. Simplification (Factoring, Rationalizing, etc.):

• If direct substitution leads to an indeterminate form like $\frac{0}{0}$, try to simplify the expression. This might involve factoring, rationalizing (multiplying by a conjugate), or canceling common terms.
• Once simplified, reapply direct substitution to find the limit.

Example: For $f(x) = \frac{x^2 - 4}{x - 2}$ as $x \to 2$: $f(x) = \frac{(x+2)(x-2)}{x-2} = x + 2$ After canceling $x - 2$, we get $\lim_{{x \to 2}} (x + 2) = 4$.

3. L'Hôpital's Rule:

• When direct substitution leads to an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, you can use L'Hôpital's Rule. This involves differentiating the numerator and the denominator separately, then taking the limit.
• You can apply L'Hôpital's Rule repeatedly if necessary.

Example: For $\lim_{{x \to 0}} \frac{\sin x}{x}$, direct substitution gives $\frac{0}{0}$. $\lim_{{x \to 0}} \frac{\sin x}{x} = \lim_{{x \to 0}} \frac{\cos x}{1} = 1$

4. Use Special Limits:

• Some limits, especially trigonometric ones, have known results. Common ones include:
  • $\lim_{{x \to 0}} \frac{\sin x}{x} = 1$
  • $\lim_{{x \to 0}} \frac{1 - \cos x}{x} = 0$
• Recognizing these can save time without further calculations.

5. Analyze One-Sided Limits:

• If the limit differs when approaching from the left ($x \to a^-$) or the right ($x \to a^+$), the limit does not exist in the usual sense. This approach is crucial for functions with discontinuities or step changes, such as piecewise functions.
• Check both one-sided limits for consistency.

Additional Tips:

• Infinite Limits: If the function grows without bound, you may encounter limits where $f(x) \to \infty$ or $f(x) \to -\infty$ as $x \to a$. This typically indicates a vertical asymptote.
• Behavior at Infinity: For limits as $x \to \infty$ or $x \to -\infty$, analyze the highest degree terms or use asymptotic behaviors.

Would you like more details on any of these techniques?

Questions to Explore Further

1. How does factoring help when finding limits in rational functions?
2. When is L'Hôpital's Rule most useful, and are there limits on its application?
3. How are one-sided limits different from regular limits?
4. What role do asymptotes play in determining the behavior of limits?
5. Why is it important to recognize indeterminate forms in limit calculations?

Tip:

Always check for indeterminate forms first, as these often guide you toward the right technique for solving limits.