# Stuck on a possibly simple trig limit

• rambo5330
In summary, the text says that the limit does not exist, but if you split the limit into two parts, it becomes 1-1.
rambo5330

## Homework Statement

how do you go about solving the equation
lim (t->0) 2t / (sin(t)) - t

the answer in the text is significantly different than what i get.. i can get most of the other trig limits using the fundamental limit etc.. but this one I am stuck ? i may be way over complicating it but i think it needs some algebraic manipulation in the current form?

## The Attempt at a Solution

You should notice that $$sin \textit{x} /\textit{x}$$ is a notable limit

lim x->0 of sinx/x =1, so
lim x->0 of 1/(sinx/x) is also 1, and 1/(sinx/x)=x/sinx, so
lim x->0 of x/sinx = 1
from here just rearrange the equation.

[STRIKE][STRIKE][/STRIKE][/STRIKE]yes I'm aware of sinx/x = 1 and all the other variations of it etc that's what i was referring to as the fundamental trig limit but i think its the fact its in the form. sinx - x i cannot factor out an x and since its in the form of a binomial this is where i think I am just doing something stupid with my algebra any pointers on that one?

rember its 2t / (sin(t)) - t

$$2t/ sin(t) - t$$

If you mean

$$\frac{2t}{\sin t -t},$$

then rewrite it in the form

$$\frac{2}{\frac{\sin t}{t} -1}.$$

If you factor out the 2 and split the limit into two parts then you have;
2 lim t->0 (t/sint) - lim t->0 (t)

that was exactly what i was doing wrong.. thank you very much

it becomes

forgive my poor latex.. I am trying

$$2 / sint/t - t/t$$
sorry ill have to figure out better latex tomorrow
but if I am correect

that is 2 / 1 - t/t and since t/t is approaching zero but both equal shouldn't that become 1 as well... giving 1-1 in the denominator and an undefined situation?? the text is saying it should equal DNE or +infinity ... i don't see the infinity situation here?

rambo5330 said:
that was exactly what i was doing wrong.. thank you very much

it becomes

forgive my poor latex.. I am trying

$$2 / sint/t - t/t$$
sorry ill have to figure out better latex tomorrow
but if I am correect

that is 2 / 1 - t/t and since t/t is approaching zero but both equal shouldn't that become 1 as well... giving 1-1 in the denominator and an undefined situation?? the text is saying it should equal DNE or +infinity ... i don't see the infinity situation here?

Well

$$\frac{1}{0}=\infty$$

in the sense that

$$\lim_{x\rightarrow 0} \frac{1}{x} \rightarrow \infty.$$

So the limit does not exist.

but in that form it just purely does not exist am i right? unless you know wether you're approaching zero from the right or the left.. you can end up with -infinity or +infinity...

so it would need to be a one sided limit.. or am i off?

You're right if the left and right limits don't agree, the limit does not exist. However, it's also true that a divergent result means that the limit doesn't exist.

In this case, the function is even, so both the right and left limits give $$-\infty$$. We still say that the limit does not exist.

excellent thanks for your help!... i did look in the textbook again to confirm the answer and i see its showing a +infinity?? not sure what the reasoning behind the positive is? but fighting with this question has actually helped me understand a bit deeper. thanks!

## 1. What is a trig limit?

A trig limit is a mathematical concept that involves finding the value of a trigonometric function as its input approaches a specific value. It is used to solve problems involving angles and triangles in calculus and other branches of mathematics.

## 2. How do I solve a trig limit?

To solve a trig limit, you need to use various trigonometric identities and properties, such as the sum, difference, double angle, and half-angle formulas. You also need to understand the behavior of trigonometric functions as their input approaches certain values, such as 0, 1, and infinity.

## 3. What are some common strategies for solving a trig limit?

Some common strategies for solving a trig limit include using algebraic manipulation, factoring, trigonometric identities, and substitution. You may also need to use L'Hopital's rule or the squeeze theorem, depending on the specific problem.

## 4. How do I know if I have simplified a trig limit correctly?

To check if you have simplified a trig limit correctly, you can use a graphing calculator to graph both the original function and the simplified function. If the two graphs overlap, then you have simplified the limit correctly.

## 5. What are some real-life applications of trigonometric limits?

Trigonometric limits are used in many fields, such as physics, engineering, and astronomy. They are used to model and predict the behavior of waves, pendulums, and other oscillating systems. They are also used in navigation and GPS technology to calculate distances and angles between objects.

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