![]() ![]() Thus, by the Intermediate Value Theorem, there are at least 2 roots for the function f(x). Similarly, there must be a root of f(x) between x = 6 and x = 8. If f(x) is a continuous function on the interval, and k is a value such that f(a) 0 > f(6). Here is the formal definition of the Intermediate Value Theorem: ![]() The Intermediate Value Theorem, often abbreviated as IVT, deals with a single function unlike the Squeeze Theorem. (Hint: use a similar tactic that we used in the first proof that is, think about the range of cos(x) and try to create bounds) The proof for the second identity is very similar. Note that this is not the only proof for the Squeeze Theorem – there is actually a more graphical version. Thus, by the Squeeze Theorem we have proven our first trigonometric identity. Since the limit as x approaches 0 of -1/x and 1/x is 0, we have: Now, we can take the limit of both sides. ![]() If we divide all parts of the inequality by x to produce the function we want to bound, we get Since we need to place upper and lower bounds on our function in order to use the Squeeze Theorem, let’s try to do that. Let’s prove the first one together using the Squeeze Theorem. It is also crucial to note that the limits of g(x) and h(x) as x approaches c must be equal to each other – if they are not equal, we cannot determine what will happen to the function between them.įrom the Squeeze Theorem, we can derive 2 important trigonometric limits: It is important that your function f(x) is always between the other 2 functions h(x) and g(x) on the interval around c, because otherwise we cannot use the “squeezing” argument on f. To put this all into cohesive statement, let’s look at a formal definition for the squeeze theorem: f(x) is squeezed between h(x) and g(x), and both h and g seem to approach a common y-value at x=c so it makes sense that f(x) would approach that same y-value at x=c as well. This theorem also has other names like the Sandwich Theorem or the Pinch Theorem, but it is most commonly called the Squeeze Theorem.Īs mentioned, the squeeze theorem can track the behavior of a function f(x), though there is one caveat: f(x) must be squeezed between 2 other functions. The squeeze theorem is a useful tool for analyzing the limit of a function at a certain point, often when other methods (such as factoring or multiplying by the conjugate) do not work. Learn about two very cool theorems in calculus using limits and graphing! The Squeeze Theorem ![]()
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