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

[karush]

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

 

[romsek]

No.

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

is the same as

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

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

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.

 

[HOI]

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

 

[karush]

MHB replies on some calculus problems