Lim (tan^2(x)/x)
x->0
Use l'Hôpital's rule:
Lim x→0 tan²(x)/x
=2sec²(x)tan(x)/1
=0/1
=0
To evaluate the limit:
lim(x->0) (tan^2(x)/x),
we can try to simplify the expression by using trigonometric identities.
Recall the identity:
tan^2(x) = sec^2(x) - 1.
Substituting this into our expression, we get:
lim(x->0) ((sec^2(x) - 1)/x).
Next, we can use another trigonometric identity:
sec^2(x) = 1 + tan^2(x).
Substituting this identity into our expression, we have:
lim(x->0) (((1 + tan^2(x)) - 1)/x).
Simplifying further, we get:
lim(x->0) (tan^2(x)/x).
Now, let's consider the numerator tan^2(x). As x approaches 0, tan(x) also approaches 0. Therefore, tan^2(x)/x also approaches 0.
Hence, the limit:
lim(x->0) (tan^2(x)/x) = 0.