Limits involving trigonometric functions

In summary, the limit of sin^2(3x)/x^2 as x tends to 0 is 9, and the limit of 2sin5x/(3x-2tan2x) as x tends to 0 is 10/3.
  • #1
Ironside
32
0

Homework Statement



1)

lim sin^2(3x)/x^2
x->0

2) lim 2sin5x/(3x-2tan2x)
x->0

Homework Equations



lim sinx/x = 1
x->0lim tanx/x = 1
x->0

The Attempt at a Solution



We just began working with limits, and we haven't covered much of trig functions at all, but our prof gave us these 2 questions to see what we can do with them. I really don't know how to begin with them.

For the first problem i tried to answer it by splitting it into 2 fractions: sin3x/x * sin3x/x then sin3x/3x * sin3x/3x * 9/1 which gave me the answer 9 for it. I'm not sure if that's right though.

I have no clue how to start on the second one. :(
 
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  • #2
The first one is perfect. For the second, I can hint you towards dividing both the denominator and the numerator by x and using the 2 limits given.
 
  • #3
bigubau said:
The first one is perfect. For the second, I can hint you towards dividing both the denominator and the numerator by x and using the 2 limits given.

I'm not following you, you mean times it by 1/x or...times x/x...?
 
  • #4
To divide something by x means to multiply it by 1/x. So you realize that

[tex] \frac{7}{6} = \frac{\frac{7}{2}}{\frac{6}{2}} [/tex]

, right ?
 
  • #5
bigubau said:
To divide something by x means to multiply it by 1/x. So you realize that

[tex] \frac{7}{6} = \frac{\frac{7}{2}}{\frac{6}{2}} [/tex]

, right ?

Ok so what i did is (2sin5x/x)/((3x-2tan2x)/x) and i separated it into 2 limits so

lim 2sin5x/x , then i do 2sin5x/5x * 5/1 then i get for the top fraction 10
x->0


then for the (3x-2tan2x)/x , i did 3x/x - 2tan2x/x , then i did 2tan2x/2x * 2/1 which gave m 4, then obviously i run into the problem that lim 3x/x when x ->0 is undefined. What did i do wrong?
 
  • #6
Ironside said:
then for the (3x-2tan2x)/x , i did 3x/x - 2tan2x/x , then i did 2tan2x/2x * 2/1 which gave m 4, then obviously i run into the problem that lim 3x/x when x ->0 is undefined. What did i do wrong?

[tex]\displaystyle\lim_{x \rightarrow 0} \frac{3x}{x} = \displaystyle\lim_{x \rightarrow 0} 3 = ?[/tex]
 
  • #7
Limit...does not exist...?
 
  • #8
Ironside said:
Limit...does not exist...?

Well, you seem to have misinterpret the word 'limit'. The function [tex]\frac{3x}{x}[/tex] is indeed undefined at x = 0.

However, when x tends to 0 (note that: x only tends to 0, i.e x is 'closed enough' to 0, not x = 0), then the function [tex]\frac{3x}{x}[/tex] has the limit of 3. This is because: [tex]\frac{3x}{x} = 3, \forall x \neq 0[/tex], so when x tends to 0, [tex]\frac{3x}{x} \rightarrow 3[/tex].

When talking about taking the limit as x tends to some value a, it means that x is near/close enough to a. The function may be undefined at x = a, but may have limit when x tends to a.
 

1. What is a limit involving trigonometric functions?

A limit involving trigonometric functions is a mathematical concept that describes the behavior of a function as its input approaches a certain value. In particular, it refers to the value that a function approaches as its input approaches a certain value, typically denoted as x → a. In the case of trigonometric functions, this value is often determined by the behavior of the function at key points on the unit circle.

2. How are trigonometric limits evaluated?

Trigonometric limits are evaluated using various techniques, including algebraic manipulation, graphing, and trigonometric identities. Some common methods for evaluating trigonometric limits include using L'Hôpital's rule, converting trigonometric functions to their equivalent exponential forms, and applying the fundamental theorem of calculus.

3. What are some common trigonometric limits?

Some common trigonometric limits include the limit of sin(x)/x as x → 0, the limit of cos(x)/x as x → 0, and the limit of tan(x)/x as x → 0. These limits are important in calculus and are often used in applications involving trigonometric functions.

4. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of a function as its input approaches a certain value from one direction (either from the left or the right). This is denoted as x → a+ (from the right) or x → a- (from the left). A two-sided limit, on the other hand, considers the behavior of a function as its input approaches a certain value from both directions. This is denoted as x → a.

5. Why are limits involving trigonometric functions important?

Limits involving trigonometric functions are important because they help us understand the behavior of these functions as their inputs approach certain values. This is essential in calculus, where limits are used to define derivatives and integrals, and in real-world applications where trigonometric functions are used to model various phenomena.

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