given f(x) = sin2x/(xcosx)
a) find lim f(x)
x->0
B) find lim f(x)
x-> pi/4
a)
Substitute x=0 into f(x) and see:
f(0)
=sin(2*0)/(0*cos(0))
=sin(0)/(0*1)
=0/0 (undefined).
To evaluate the limit, you could use d'Hôpital's rule or make use of the known limit of sin(x)/x=1 to get
lim x->0 sin(2x)/(xcos(x))=2
b)
Substitute x=π/4
f(π/4)=sin(2*π/4)/((π/4)*cos(π/4))
=1/((π/4)*(√2/2))
=4√2 / π
To find the limit of a function as x approaches a specific value, you can directly substitute that value into the function and simplify.
a) Let's find the limit as x approaches 0.
lim(f(x)) as x approaches 0 = f(0)
To find f(0), substitute x = 0 into the given function f(x):
f(0) = sin^2(0) / (0 * cos(0))
= 0 / (0 * 1)
= 0 / 0
Here, we obtained an indeterminate form, 0/0. This means we cannot directly evaluate the limit using direct substitution.
To proceed, we can simplify the function using trigonometric identities. Notice that sin(0) = 0 and cos(0) = 1, so we have:
f(0) = 0 / (0 * 1)
= 0 / 0
We can rewrite the function using these trigonometric identities:
f(x) = sin^2(x) / (x * cos(x))
= (sin(x))^2 / (x * cos(x))
= (sin(x) / x) * (sin(x) / cos(x))
Now, we have two separate limits to find:
lim(sin(x) / x) as x approaches 0
lim(sin(x) / cos(x)) as x approaches 0
To evaluate these limits, we can use the Squeeze Theorem, where we know that -1 ≤ sin(x) ≤ 1 and -1 ≤ cos(x) ≤ 1 for all x. It follows that:
-1 ≤ sin(x) / x ≤ 1
-1 ≤ sin(x) / cos(x) ≤ 1
Since the limits of -1 and 1 as x approaches 0 are both 1, we can conclude that:
lim(sin(x) / x) as x approaches 0 = 1
lim(sin(x) / cos(x)) as x approaches 0 = 1
Therefore, the limit of f(x) as x approaches 0 is:
lim(f(x)) as x approaches 0 = f(0)
= (sin(0) / 0) * (sin(0) / cos(0))
= 1 * 1
= 1
b) Let's find the limit as x approaches π/4.
lim(f(x)) as x approaches π/4 = f(π/4)
To find f(π/4), substitute x = π/4 into the given function f(x):
f(π/4) = sin^2(π/4) / (π/4 * cos(π/4))
= (1/2)^2 / (π/4 * (√2/2))
= 1/4 / (π/4 * (√2/2))
= (√2 / 4π) / (√2 / 2)
= (√2 / 4π) * (2 / √2)
= 1 / (2π)
Therefore, the limit of f(x) as x approaches π/4 is:
lim(f(x)) as x approaches π/4 = f(π/4)
= 1 / (2π)