Lim Tan^2(3x)/2x^2
X-0
Find the Limit?
To find the limit of the expression lim(x->0) (tan^2(3x))/(2x^2), we can apply the limit properties and trigonometric identities. Let's go through the steps:
Step 1: Substitute x = 0 into the expression:
lim(x->0) (tan^2(3x))/(2x^2) = (tan^2(3*0))/(2*0^2) = tan^2(0)/0
Step 2: Simplify the expression:
tan^2(0) is equal to 0, so we have:
lim(x->0) (tan^2(3x))/(2x^2) = 0/0
Step 3: We have the indeterminate form 0/0, which means we cannot directly evaluate the limit at this point. We need to apply additional tools such as L'Hôpital's rule or trigonometric identities to simplify the expression.
Using the trigonometric identity: tan^2(x) = sec^2(x) - 1, we can rewrite the expression:
lim(x->0) ((sec^2(3x) - 1))/(2x^2)
Step 4: Now we can substitute x = 0 into the expression:
lim(x->0) ((sec^2(3*0) - 1))/(2*0^2) = ((sec^2(0) - 1))/0
Step 5: Simplify the expression:
sec^2(0) is equal to 1, so we have:
lim(x->0) ((1 - 1))/0 = 0/0
Step 6: We still have an indeterminate form, so we need to apply further simplification.
Using another trigonometric identity: sec^2(x) = 1 + tan^2(x), we can rewrite the expression:
lim(x->0) ((1 + tan^2(0) - 1))/0 = lim(x->0) (tan^2(0))/0
Step 7: Substitute tan^2(0) = 0 into the expression:
lim(x->0) (0)/0 = 0/0
Step 8: We are still left with an indeterminate form, so we need to apply a different approach. In this case, we can use the fact that tan(x) is equivalent to sin(x)/cos(x) and rewrite the expression as:
lim(x->0) (sin^2(0)/cos^2(0))/(2x^2) = lim(x->0) (sin^2(0))/(cos^2(0) * 2x^2)
Step 9: Substitute sin^2(0) = 0 and cos^2(0) = 1 into the expression:
lim(x->0) (0)/(1 * 2x^2) = 0/0
Step 10: We still have the indeterminate form 0/0. To resolve this, we can simplify the expression further by canceling out common factors. In this case, the x^2 term in the denominator can be canceled out with one factor of x in the numerator.
lim(x->0) (0)/(2x^2) = 0
Hence, the limit of the expression lim(x->0) (tan^2(3x))/(2x^2) is equal to 0.