Given #(sin^2(x^2))/(x^4)# how do you find the limit as x approaches 0?

Question

Paisley Johnson · Accepted Answer

1

we'll use well known limit #= lim_{z to 0} (sin(z))/(z) = 1# #lim_{x to 0} (sin^2(x^2))/(x^4)# #= lim_{x to 0} (sin(x^2))/(x^2) (sin(x^2))/(x^2)# let # y = x^2# #= lim_{y to 0} (sin(y))/(y) (sin(y))/(y)# because both limits exist, as stated above, we can separate them out #= lim_{y to 0} (sin(y))/(y) * lim_{y to 0} (sin(y))/(y)# #= 1 * 1 = 1# Link to Basic Limit Laws

Cameron Alexander · Answer

undefined To find the limit as x approaches 0 for the expression (sin^2(x^2))/(x^4), we can use the limit properties and trigonometric identities.

First, we simplify the expression by applying the identity sin^2(x) = (1 - cos(2x))/2:

(sin^2(x^2))/(x^4) = ((1 - cos(2x^2))/2)/(x^4)

Next, we simplify further by multiplying the numerator and denominator by 2:

((1 - cos(2x^2))/(2x^4))

Now, we can evaluate the limit as x approaches 0.

By applying the limit properties, we can take the limit of each term separately:

lim(x->0) (1 - cos(2x^2)) = 1 - cos(0) = 1 - 1 = 0

lim(x->0) (2x^4) = 2(0)^4 = 0

Finally, we divide the limits of the numerator and denominator:

lim(x->0) ((1 - cos(2x^2))/(2x^4)) = 0/0

Since we have an indeterminate form of 0/0, we can apply L'Hôpital's rule.

Differentiating the numerator and denominator with respect to x:

lim(x->0) (d/dx(1 - cos(2x^2)))/(d/dx(2x^4))

Applying the chain rule and simplifying:

lim(x->0) (4xsin(2x^2))/(8x^3)

Simplifying further:

lim(x->0) (sin(2x^2))/(2x^2)

Now, we can evaluate the limit again:

lim(x->0) (sin(2x^2))/(2x^2) = sin(2(0)^2)/(2(0)^2) = sin(0)/0 = 0/0

Again, we have an indeterminate form of 0/0. Applying L'Hôpital's rule once more:

Differentiating the numerator and denominator with respect to x:

lim(x->0) (d/dx(sin(2x^2)))/(d/dx(2x^2))

Applying the chain rule and simplifying:

lim(x->0) (4xcos(2x^2))/(4x)

Simplifying further:

lim(x->0) cos(2x^2) = cos(2(0)^2) = cos(0) = 1

Therefore, the limit as x approaches 0 for the expression (sin^2(x^2))/(x^4) is 1.