Need help asap! what is the limit as x approaches 0 of -x/tan(3x) ?
You should have either proved or collected several basic trig limits
e.g.
limit sinx/x as x---> 0 = 1
also
lim tanØ/Ø as Ø ---> 0 = 1 which implies lim Ø/tanØ as Ø ---> 0 = 1
so let's make your question look like that
lim -x/tan(3x)
= lim -3x/(3tan(3x))
= lim (-1/3) [(3x)/tan(3x)] , (where 3x = Ø)
= lim (-1/3) [Ø/tanØ]
= (-1/3)(1)
= -1/3
To find the limit as x approaches 0 of -x/tan(3x), we can use basic properties of limits and trigonometric identities. Here's how we can proceed:
Step 1: Identify any removable discontinuity.
In this case, the function is not defined at x = 0 since the denominator becomes zero when x = 0. However, we can simplify the expression using trigonometric identities to eliminate this discontinuity.
Step 2: Apply the trigonometric identity.
The tangent function can be rewritten as sin(x)/cos(x), so we have:
-x / (sin(3x) / cos(3x))
Step 3: Simplify the expression.
We can simplify this expression further using the properties of fractions:
- x * cos(3x) / sin(3x)
Step 4: Evaluate the limit.
Now that we have a simplified expression, we can find the limit by substituting x = 0 into the expression:
lim(x->0) [-x * cos(3x) / sin(3x)]
As x approaches 0, the numerator -x and cos(3x) approach 0, while the denominator sin(3x) approaches 0 as well. This results in an indeterminate form of 0/0.
Step 5: Apply L'Hôpital's Rule (if necessary).
To resolve this indeterminate form, we can apply L'Hôpital's Rule. This rule states that if both the numerator and denominator of a fraction approach 0 (or ±∞) as x approaches a certain value, the limit of their quotient can often be found by taking the derivative of the numerator and the denominator separately.
Differentiating the numerator and denominator with respect to x, we have:
lim(x->0) [(-cos(3x) - 3x * (-sin(3x))) / (3cos^2(3x))]
Step 6: Simplify the expression (continued).
Now, evaluate the expression after applying L'Hôpital's Rule:
lim(x->0) [(-cos(3x) + 3x * sin(3x)) / (3cos^2(3x))]
Step 7: Evaluate the limit (continued).
Substituting x = 0 into the expression, we have:
(-cos(0) + 0 * sin(0)) / (3cos^2(0))
(-1 + 0) / (3 * 1)
-1/3
Therefore, the limit as x approaches 0 of -x/tan(3x) is -1/3.