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Evaluate lim sin(x)/x as x approaches 0 - a foundational calculus limit
Key values: sin(x)/x · x -> 0 · Result: 1
Explore the limit of (1 + 1/x)^x as x approaches infinity (Euler's number)
Key values: (1+1/x)^x · x -> Infinity · Result: e
Evaluate the derivative definition limit for x² at x = 3
Key values: (x² - 9)/(x - 3) · x -> 3 · Result: 6
Evaluating the behavior of as reveals the function's end behavior and identifies horizontal asymptotes.
A function can have different horizontal asymptotes as and .
For where has degree and has degree :
Dominant term technique: Divide numerator and denominator by the highest power of in the denominator. All terms with in the denominator go to zero as .
The growth hierarchy: logarithms grow slower than polynomials, which grow slower than exponentials. Formally: for any and .
A limit at infinity describes the behavior of as grows without bound (toward positive or negative infinity). If the limit equals a finite value , then is a horizontal asymptote of the function.
Compare the degrees of the numerator () and denominator (). If , the limit is 0. If , the limit is the ratio of leading coefficients. If , the limit is and there is no horizontal asymptote.
Divide the numerator and denominator by the highest power of in the denominator. As , all terms with in the denominator approach zero, leaving only the dominant terms that determine the limit.
Yes. A function can approach different values as versus . For example, approaches 1 as and as .
Logarithms grow slower than any positive power of , which grows slower than exponential functions. Formally: for any and . This hierarchy determines which term dominates at infinity.