What is the limit as t approaches 0 for (sint)/(1+cost)?
To find the limit as t approaches 0 for (sin t) / (1 + cos t), we can use some trigonometric identities and L'Hôpital's rule.
First, let's simplify the expression. We can rewrite sin t / (1 + cos t) as (sin t / cos t) / ((1 / cos t) + (cos t / cos t)). Simplifying further, we get:
= tan t / (sec t + 1).
Now, let's find the limit as t approaches 0:
lim t->0 (tan t / (sec t + 1)).
Since the expression includes tangent, we can rewrite it as sine over cosine:
= lim t->0 (sin t / cos t) / (sec t + 1).
Now, we can apply L'Hôpital's rule, which states that if we have an indeterminate limit of the form 0/0 or ∞/∞, we can take the derivative of the numerator and denominator and then evaluate the limit again.
Let's differentiate the numerator and denominator:
Numerator: d/dt(sin t) = cos t,
Denominator: d/dt(sec t + 1) = sec t * tan t.
Now, let's find the limit of the differentiated expression:
= lim t->0 (cos t / (sec t * tan t)).
As t approaches 0, cos t approaches 1, sec t approaches 1, and tan t approaches 0. So we have:
= 1 / (1 * 0) = 1 / 0.
Since the denominator is approaching 0, the limit is undefined.
Therefore, the limit as t approaches 0 for (sin t) / (1 + cos t) is undefined.