How to find LIMITS of functions?

Question

Please explain how you got to the answer.

Any help would be appreciated!

Find the limit:

1) lim as x approaches 0 of (x / sin(3x))

How do I remove the zero from the denominator?

2) lim as x appr. 2 from the right of (sq.rt. (2x-1))

3) lim as x appr. 3 of (3/(x^2 - 6x +9))

Again, how do I get rid of the zeros in the denom? The bottom polynomial factors out to (x-3)^2, but how do I go further?

4) lim as x appr. 0 of (3x + 2+ (1/x^2))

I put it together into one fraction ((3x^3 + 2x^2 + 1)/x^2) but I don't know if that will help at all.

Thank you for any help you can offer!

Answer 1

1) There's a famous limit that's proved geometrically that goes by this form:

lim sin(x) / x = 1

x->0

We can use the algebra of limits to adapt this limit to your situation.

lim x / sin(x) = 1

x->0

lim 3x / sin(3x) = 1

x->0

lim x / sin(3x) = 1/3

x->0

2)lim sqrt(2x - 1)

x->2+

sqrt(2x - 1) is a continuous function, and so the limit will be the value of the function at x = 2, which will be sqrt(3), so:

lim sqrt(2x - 1) = sqrt(3)

x->2+

3) The 3 in the numerator stays the same, while the denominator becomes very small and the entire expression becomes very large. The limit is positive infinity. Notice that both sides of 3 approach positive infinity, which is important, otherwise the limit wouldn't exist.

4) It won't really help. The 3x + 2 part will just approach a finite number (i.e. 2), but the 1/x^2 term will get larger and larger, and will bring the limit up to positive infinity.

Answer 2

1.

Lim x->0 (x/Sin(3x))

Use LHospitals Theorem

= Lim x->0 (1/3Cos(3x) = 1/3

Answer 3

1. multiply and divide by 3 so tat lt of 3x/sin3x becomes 1 and you   get 1/3 as ans.

2. i think you must directly apply the value.

3. try out L' Hospital rule.

4. same as 2.  

Answer 4

1) According to Hospital RULE,Lim x/sin3x when xapproaches zero,

you can say F'x/f ' Sin 3x.

Then 1/3Cos 3x = 1/3X1 = 1/3.

Answer 5

This type of  problems where   limit becomes 0/0 can be  solved by  L'Hospital rule which states

Lt -- 0 f(X) / g(x)  = 0/0 then  limit is f '(x) /g '(x).

Hence in all problems  take the derivative  and then set X  = 0 to find the value

Answer 6

1) Use L'Hospital's rule, getting:

= lim x --> 0 [1/3cos(3x)] = 1/3

2) Just plug in 2 and get sqrt(3)

3) No limit. f(x) increases without limit.

4) No limit. Same as problem 3