Limit Properties and Finding Strategies

Limits describe what happens to a function as its input approaches a specific value. The key limit properties you need to know include addition, subtraction, multiplication, division, exponentiation, and composition. For example, the limit of a sum equals the sum of the limits, and the limit of a product equals the product of the limits.

When finding limits, always start with direct substitution. If substitution gives you an actual value, you're done! If you get ∞/0, that indicates an asymptote. But if you get the indeterminate form 0/0, you'll need to try other techniques.

The indeterminate form 0/0 doesn't mean the limit doesn't exist - it just means you need more work to find it. This is where special techniques come into play.

Quick Tip: When you encounter 0/0, think of it as a signal to try factoring, conjugates, or trig identities - not as a final answer!

Continuity and Special Techniques

When stuck with indeterminate forms, try factoring polynomials, using conjugates for square roots, applying trig identities, or as a last resort, creating tables or graphs to approximate the limit. Each technique works best for specific types of expressions.

Functions can have three main types of discontinuities: removable (point) discontinuities that can be "fixed," jump discontinuities where the function suddenly changes value, and asymptotic discontinuities where the function grows infinitely large. A function is continuous at a point if the limit at that point equals the function's value.

The Squeeze Theorem is a powerful tool when direct methods fail. If a function g(x) is "squeezed" between two functions f(x) and h(x) that both approach the same limit L, then g(x) must also approach L.

Remember these crucial limits: sin(θ)/θ → 1 and $1-cos(θ)$/θ → 0 as θ approaches 0. These special limits appear frequently in calculus problems.

Infinite Limits and Intermediate Value Theorem

When a limit approaches infinity as x approaches some value a, we have a vertical asymptote at x = a. For rational functions with polynomials, you can determine the limit by comparing the highest powers of x in the numerator and denominator.

For limits of the form Axᵃ/Bxᵇ as x approaches infinity, the result depends on comparing exponents a and b: if a = b, the limit equals A/B; if a < b, the limit is 0; if a > b, the limit is infinity. This technique helps evaluate many complicated limits involving polynomials.

The Intermediate Value Theorem guarantees that continuous functions don't skip values. If a function is continuous on an interval [a,b], then it takes on every value between f(a) and f(b). This powerful theorem helps us prove that certain equations have solutions.

Why This Matters: The Intermediate Value Theorem explains why you can't drive from sea level to a mountain without passing through every elevation in between - continuous functions can't "teleport" past values!