I do not understand this:
Find the limit as theta approaches 0 for (cos theta)/ x
hint: apply L'Hopital's Rule
math(dot)hmc(dot)edu/calculus/tutorials/lhopital/
To find the limit as theta approaches 0 for (cos theta)/x, we can use the limit properties and trigonometric identities.
First, let's rewrite the expression:
(cos theta)/x = cos theta * (1/x)
Now, we can approach this limit in two steps:
Step 1: Evaluate the limit of cos theta as theta approaches 0.
The limit of cos theta as theta approaches 0 is equal to cos(0), which is 1. This is a standard trigonometric limit.
Step 2: Evaluate the limit of (1/x) as x approaches 0.
The limit of (1/x) as x approaches 0 can be determined by analyzing the behavior of 1/x as x gets arbitrarily close to 0.
If we approach x = 0 from the right side (x > 0), the values of 1/x will tend to positive infinity (1/x → +∞).
On the other hand, if we approach x = 0 from the left side (x < 0), the values of 1/x will tend to negative infinity (1/x → -∞).
Since theta approaches 0 but x approaches 0 from either side, the limit of (1/x) as x approaches 0 does not exist.
Now, to find the overall limit, we multiply the limits obtained in step 1 and step 2:
Limit = (cos theta) * (1/x) = 1 * (1/x) = 1/x
Therefore, the limit as theta approaches 0 for (cos theta)/x is simply 1/x.