Conjecturing the Limit and Finding Delta for Sinusoidal Function

[John O' Meara]

Let $$f(X)=\frac{\sin(2x)}{x}$$ and use a graphing utility to conjecture the value of L = $$\lim_{x->0}f(x) \mbox{ then let } \epsilon =.1$$ and use the graphing utility and its trace feature to find a positive number $$\delta$$ such that $$|f(x)-l|< \epsilon \mbox{ if } 0 < |x| < \delta$$. My conjecture of the limit L = 2, therefore if that is the case then $$1.9< f(x) < 2.1$$. Since the maximum value of f(x) < 2, the graphing utility will not be able to find delta will it? What is the value of delta if L=2?Thanks.

 

[LeonhardEuler]

It will be able to. The function does not need to go both above and below. You just need to find a number $$\delta$$ so that f(x) is between those numbers whenever x is in the interval $$[-\delta,\delta]$$.

 

[John O' Meara]

I set up my graphing calculator Xmin =.2758..., Xmax = .2775... Then f(x)=y=1.8999954 gives x_0=.27596197. The length 0 to x_0 is not = delta but this half interval has the property that for each x in the interval (except possibly for x=0) the values of f(x) is between either 0 and L+epsilon or L-epsilon. delta could be much smaller than x_0. I still have not found delta.

 

[LeonhardEuler]

Any positive number delta that has that property ("for each x in the interval (except possibly for x=0) the values of f(x) is between either 0 and L+epsilon or L-epsilon") is a solution to the problem as long as it does not specifically ask for the largest possible number delta with that property.