How do you find #lim (3x^2+x+2)/(x-4)# as #x->0# using l'Hospital's Rule or otherwise?

Answer 1

#-1/2#

The initial form is not indeterminate. We cannot use l'Hospiital for this.

#lim_(xrarr0)(3x^2+x+2) = 2# and
#lim_(xrarr0)(x-4) = -4#, so
#lim_(xrarr0)(3x^2+x+2)/(x-4) = 2/(-4) = -1/2#
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

To find ( \lim_{x \to 0} \frac{3x^2 + x + 2}{x - 4} ), we first try to directly substitute ( x = 0 ) into the expression. However, this results in an indeterminate form of (\frac{2}{-4}). Therefore, we can apply L'Hôpital's Rule by taking the derivatives of the numerator and denominator with respect to ( x ) separately until we get a determinate form.

[ f(x) = 3x^2 + x + 2 ] [ g(x) = x - 4 ]

Taking the derivatives: [ f'(x) = 6x + 1 ] [ g'(x) = 1 ]

Now, applying L'Hôpital's Rule: [ \lim_{x \to 0} \frac{3x^2 + x + 2}{x - 4} = \lim_{x \to 0} \frac{6x + 1}{1} = \frac{1}{1} = 1 ]

So, ( \lim_{x \to 0} \frac{3x^2 + x + 2}{x - 4} = 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