How do you find the limit of # (cos(x)/sin(x) + 1) # as x approaches 0 using l'hospital's rule?

Answer 1

See below.

Plugging in zero gives:

#1/0 + 1#

L'Hospital's Rule can only be used when we have a quotient of an indeterminate form:

#0/0# , #oo/oo#

Since we do not have this form, l'hospital's rule cannot be used.

If you evaluate the limit by plugging in values approaching 0 from the left the limit will be:

#lim_(x->0^-)cos(x)/sin(x) +1= -oo#

And from the right:

#lim_(x->0^+)cos(x)/sin(x) +1= oo#

So the limit is undefined

Using L'Hospital's rule:

#d/dx cos(x) = -sin(x)#
#d/dx sin(x) = cos(x)#
#lim_(x->0)((-sin(x))/cos(x) + 1)= ((-sin(0))/cos(0)) +1 =( 0/1 + 1)=color(red)(1)#
#color(red)(lim_(x->0^)cos(x)/sin(x) +1 != 1)#

So L'Hospital's rule fails.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2
#lim_(x->0) (frac{cosx}{sinx}+1)#
# = lim_(x->0) frac{cosx+sinx}{sinx}#
By direct substitution, this gives #1/0#. This is not one of the indeterminate forms of L'Hospital's rule.
#lim_(x->0^-) frac{cosx+sinx}{sinx} = -oo#
#lim_(x->0^+) frac{cosx+sinx}{sinx} = oo#
#:. lim_(x->0) frac{cosx+sinx}{sinx} " DNE"#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To find the limit of (cos(x)/sin(x) + 1) as x approaches 0 using L'Hospital's rule, we can differentiate the numerator and denominator separately.

Differentiating the numerator, we get -sin(x). Differentiating the denominator, we get cos(x).

Now, we can evaluate the limit of the derivative of the numerator divided by the derivative of the denominator as x approaches 0.

The derivative of the numerator (-sin(x)) evaluated at x = 0 is -sin(0) = 0. The derivative of the denominator (cos(x)) evaluated at x = 0 is cos(0) = 1.

Therefore, the limit of (cos(x)/sin(x) + 1) as x approaches 0 using L'Hospital's rule is 0/1 = 0.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 4

To find the limit of ( \frac{\cos(x)}{\sin(x)} + 1 ) as ( x ) approaches 0 using L'Hospital's Rule, follow these steps:

  1. Evaluate the expression at the limit point: ( \frac{\cos(0)}{\sin(0)} + 1 = \frac{1}{0} + 1 )

  2. Since ( \frac{1}{0} ) is undefined, we need to rewrite the expression to apply L'Hospital's Rule. Rewrite ( \frac{\cos(x)}{\sin(x)} ) as ( \frac{\frac{d}{dx}\cos(x)}{\frac{d}{dx}\sin(x)} ).

  3. Differentiate the numerator and denominator separately with respect to ( x ):

    • Derivative of ( \cos(x) ) is ( -\sin(x) ).
    • Derivative of ( \sin(x) ) is ( \cos(x) ).
  4. Substitute the derivatives back into the expression: ( \frac{-\sin(x)}{\cos(x)} )

  5. Evaluate the expression at the limit point: ( \frac{-\sin(0)}{\cos(0)} = \frac{0}{1} = 0 )

  6. Add 1 to the result: ( 0 + 1 = 1 )

Therefore, the limit of ( \frac{\cos(x)}{\sin(x)} + 1 ) as ( x ) approaches 0 is 1.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7