How do you find the lim as x->infinity of (x+sin(x))/(x+cos(x))??
divide numerator and denominator by x
lim (1+sinx/x)/(1+cosx/x) which is 1
I don't understand how you got from lim (1+sinx/x)/(1+cosx/x) to the answer. I know that sinx/x is 1 so would the top be 2?
NO. lim sinx/x when x=1 approaches zero, but this is lim x>>inf so lim is zero.
Thank you.
To find the limit as x approaches infinity of the given function (x+sin(x))/(x+cos(x)), we can analyze the ratio of the highest degree terms of the numerator and the denominator.
In this case, the highest degree term in both the numerator and the denominator is x, since sin(x) and cos(x) are bounded functions.
Thus, we can divide both the numerator and the denominator by x to simplify the expression. This gives us:
lim(x->infinity) (x+sin(x))/(x+cos(x)) = lim(x->infinity) (1+sin(x)/x)/(1+cos(x)/x)
Now, as x approaches infinity, both sin(x)/x and cos(x)/x tend to zero. This is a well-known result from calculus, but if you're unfamiliar with it, you can verify it by applying the limit definition of sin(x)/x and cos(x)/x.
Therefore, we have:
lim(x->infinity) (1+sin(x)/x) / (1+cos(x)/x) = (1+0) / (1+0) = 1
Hence, the limit as x approaches infinity of (x+sin(x))/(x+cos(x)) is equal to 1.