# Find Limit of $\displaystyle\frac{\sec x +3}{7x-\tan y}$ at (0,$\dfrac{\pi}{4}$)

• MHB
• karush
In summary, the conversation discusses finding the limit of a function represented by $\dfrac{\sec x + 3}{7x - \tan y}$ at the point (0, π/4). It is noted that this can be written as finding the limit as x approaches 0 and y approaches π/4. The speaker mentions that plugging in these values may result in difficulties, but in this case, the limit value can be obtained directly by plugging in the values. It is also mentioned that the limit is equivalent to the function value if the function is continuous at the given point.
karush
Gold Member
MHB
Find the limit
$\displaystyle\lim_{(x,y) \to \left[0,\dfrac{\pi}{4}\right]} \dfrac{\sec x +3}{7x-\tan y}=$

I haven't seen limit displayed like this so assume the (x,y) values are just pluged in as first step

Last edited:
No.

$$\displaystyle (x,y) \to \left [0, \dfrac \pi 4\right]$$

is the same as

$$\displaystyle x \to 0,~y \to \dfrac \pi 4$$

i.e the limit at the point $$\displaystyle \left(0,~ \dfrac \pi 4\right)$$

If you can plug them in and get a value, great, that's your limit.
But generally plugging the values in will result in 0 in the denominator, or infinity divided by infinity, or
any of the usual difficulties one encounters doing limit problems.

In this particular problem you can just plug the values in and obtain the limit value directly.

$\lim{x\to a} f(x)$ is the same as f(a) if and only if f is continuous at x= a. Indeed that is the definition of "continuous".

## 1. What is the definition of a limit?

A limit is a mathematical concept that describes the behavior of a function as the input approaches a specific value. It is used to determine the value that a function approaches, or "approaches," as the input gets closer and closer to a certain point.

## 2. How do you find the limit of a function?

To find the limit of a function, you need to evaluate the function at the specific value or point where the limit is being taken. This can be done by plugging in values that are closer and closer to the specific point until you can determine the value that the function is approaching.

## 3. What is the limit of a function at a specific point?

The limit of a function at a specific point is the value that the function approaches as the input gets closer and closer to that point. It is not necessarily the same as the value of the function at that point.

## 4. Can the limit of a function exist at a point where the function is not defined?

Yes, the limit of a function can exist at a point where the function is not defined. This is because the limit is determined by the behavior of the function as the input approaches the specific point, not the actual value of the function at that point.

## 5. How do you find the limit of a complex function, such as $\displaystyle\frac{\sec x +3}{7x-\tan y}$?

To find the limit of a complex function, such as $\displaystyle\frac{\sec x +3}{7x-\tan y}$, you need to follow the same steps as finding the limit of a simpler function. First, evaluate the function at the specific point by plugging in values that are closer and closer to the point. If the function is indeterminate at that point, you may need to use algebraic manipulation or other techniques to simplify the function and determine the limit.

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