Limits & Continuity: Key Concepts and Techniques

Ever wondered what happens when you approach a point on a graph but never quite reach it? That's what limits help us understand! When working with limits, we have several techniques to find their values.

Direct substitution is the simplest approach - just plug in the value. For example, $\lim_{x \to -1} (6x^2 + 5x - 1) = 6(-1)^2 + 5(-1) - 1 = 6 - 5 - 1 = 0$. Remember, a function is continuous at point $a$ if $\lim_{x \to a} f(x) = f(a)$.

With composite functions, we can use the theorem: $\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x))$, but only when $\lim_{x \to a} g(x)$ exists and $f$ is continuous at that limit. For trigonometric functions, we apply direct substitution: $\lim_{x \to \pi} \sin(x) = \sin(\pi) = 0$.

Piecewise functions require checking the appropriate piece based on where $a$ falls. For example, with $f(x) = \begin{cases}x + 2 & \text{for } 0 \leq x \leq 4\\sqrt{x} & \text{for } x > 4\end{cases}$, you need to evaluate the limit from the left and right at transition points to determine if the overall limit exists.

Pro Tip: When facing a rational function with a zero in the denominator, try factoring first! Often, you can cancel terms to find a simplified expression valid everywhere except at that problematic point.

Sometimes limits don't exist (DNE), especially when approaching from different directions gives different results, or when we end up with indeterminate forms like $\frac{0}{0}$.