Question

True or False The limit of a quotient equals the quotient of the limits.

Short answer

The given statement is False.

Step-by-step solution

Step 1. Given Information

The statement given is, "The limit of a quotient equals the quotient of the limits.

Step 2. Example 

Let us take for example,

$f\left(x\right)=\frac{x^2-x-6}{x^2-9}$

$\frac{x^2-x-6}{x^2-9}=\frac{(x-3)(x+2)}{(x-3)(x+3)}$

On Applying the limit,

$\underset{x\to 3}{\lim }\frac{(x-3)(x+2)}{(x-3)(x+3)}$

The limit of the quotient of the function$\underset{x\to 3}{\lim }\frac{1}{x^2-9}=\frac{1}{0}$which is undefined

Step 3. Conclusion

This statement is false because if in any case the limit of the denominator becomes zero, Then the limit is undefined.