AP Calculus AB 1.8 Squeeze Theorem Example with "e" and a Polynomial

Please subscribe!    / nickperich   The Squeeze Theorem is a useful tool in calculus to evaluate the limit of a function when the function itself is difficult to analyze directly. It works by "squeezing" the function of interest between two other functions that have the same limit at a certain point. *The Squeeze Theorem states:* If three functions \( f(x) \), \( g(x) \), and \( h(x) \) satisfy: \[ g(x) \leq f(x) \leq h(x) \] for all \( x \) in some interval around \( c \) (except possibly at \( c \) itself), and if: \[ \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L, \] then: \[ \lim_{x \to c} f(x) = L. \] Example: Suppose you want to find the limit: \[ \lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right). \] Directly finding this limit can be tricky because \( \sin\left(\frac{1}{x}\right) \) oscillates wildly as \( x \) approaches 0. However, we know that: \[ -1 \leq \sin\left(\frac{1}{x}\right) \leq 1. \] By multiplying all parts of this inequality by \( x^2 \) (which is non-negative for small \( x \)), we get: \[ -x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2. \] Now, the limits of the outer functions as \( x \to 0 \) are: \[ \lim_{x \to 0} -x^2 = 0 \quad \text{and} \quad \lim_{x \to 0} x^2 = 0. \] By the Squeeze Theorem, since \( -x^2 \) and \( x^2 \) both approach 0 as \( x \to 0 \), we conclude: \[ \lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0. \] The Squeeze Theorem helped us find this limit by bounding the function between two simpler functions. I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study