Use a graph to estimate:
Lim x-> 0 of sin(8x)/x
when x is measured in degrees.
To estimate the limit of sin(8x)/x as x approaches 0 when x is measured in degrees, we can use a graph.
1. Start by plotting the graph of y = sin(8x)/x.
2. Choose a range of x-values around 0, for example, from -10 to 10.
3. Calculate the y-values for each x-value using the formula y = sin(8x)/x.
4. Plot the points (x, y) on the graph.
5. As x approaches 0, observe the behavior of the graph.
6. Estimate the limit by looking at the trend of the y-values as x gets closer to 0.
Here's how you can estimate the limit using a graph:
- Draw the x and y axes on a graph paper.
- Label the x-axis as "x (in degrees)" and the y-axis as "y".
- Choose a suitable scale for both axes.
- Plot the points for different x-values, calculating the corresponding y-values using the formula y = sin(8x)/x.
- Connect the points on the graph to visualize the shape of the function.
Now, as x approaches 0, observe the behavior of the graph. If you find that the y-values tend to approach a specific value, you can estimate the limit to be that value. However, if the y-values become unbounded (approach infinity or negative infinity) or do not approach a specific value, the limit does not exist.
By analyzing the graph, you will get an estimate for the limit of sin(8x)/x as x approaches 0 when x is measured in degrees.
change it to radians.
lim sin(8*2PI/360 x)/(2PI/360 x) x in radians, lim as x>>>0
lim sin(16PI/360 x)/x * 360/2PI
using L'Hopitals rule, to make certain..
lim= cos(16PI/360 x) * 16PI/360 /1 * 360/2PI
= 1*16PI*360/360*2PI= 8
another way to do it. Let u=8x
du=8dx
lim sin u/u/8= 8