Evaluate the limits without using a calculator. Justify all steps in the solutions.
lim x-->0 x^7cos(3/x)
the max cosine can ever be is -1 or 1, it is always between that, so cos (inf) is from -1 to 1. That is a finite number, multiplying it by 0^7 is zero, at least here in TExas
14-2.6+2
To evaluate the limit lim x→0 x^7cos(3/x), we can use the fact that the cosine function is bounded between -1 and 1. This means that cos(3/x) is always between -1 and 1, regardless of the value of x.
Now, let's analyze the expression x^7cos(3/x) as x approaches 0. We know that as x approaches 0, x^7 becomes very close to 0. And since cos(3/x) is bounded between -1 and 1, multiplying it by x^7 will not change the fact that the overall expression will approach 0.
Therefore, the limit lim x→0 x^7cos(3/x) is equal to 0. We have justified this by considering the behavior of x^7 as x approaches 0 and the fact that cos(3/x) is bounde