Question

In Exercises 49 and 50 , determine the limit. Assume that $$\lim _ { x \rightarrow b } f ( x ) = 7$$ and $$\lim _ { x \rightarrow b } g ( x ) = - 3$$ (a) $$\lim _ { x \rightarrow b } ( f ( x ) + g ( x ) ) \quad \quad$$ (b) $$\lim _ { x \rightarrow b } ( f ( x ) \cdot g ( x ) )$$ (c) $$\lim _ { x \rightarrow b } 4 g ( x ) - \quad$$ (d) $$\lim _ { x \rightarrow b } \frac { f ( x ) } { g ( x ) }$$

Short answer

(a) The limit is 4. (b) The limit is -21. (c) The limit is -12. (d) The limit is \(-\frac{7}{3}\).

Step-by-step solution

Find the limit of (f(x) + g(x))

The sum of the limits of two functions is equal to the limit of their sum. Therefore, we substitute the given limits into the sum, like so: \(\lim _ { x \rightarrow b } ( f ( x ) + g ( x ) ) = \lim _ { x \rightarrow b } f ( x ) + \lim _ { x \rightarrow b } g ( x ) = 7 - 3 = 4\)

Find the limit of (f(x) * g(x))

The limit of the product of two functions is equal to the product of their limits. We substitute the given limits into the sum like so: \(\lim _ { x \rightarrow b } ( f ( x ) \cdot g ( x ) ) = \lim _ { x \rightarrow b } f ( x ) \cdot \lim _ { x \rightarrow b } g ( x ) = 7 \cdot -3 = -21\)

Find the limit of 4g(x)

The limit of a constant times a function is equal to the constant times the limit of the function. We substitute the given limit of g(x) into the equation like: \(\lim _ { x \rightarrow b } 4 g ( x ) = 4 \cdot \lim _ { x \rightarrow b } g ( x ) = 4 \cdot -3 = -12\)

Find the limit of (f(x) / g(x))

The limit of the division of two functions is equal to the division of the limit of the functions, provided that the limit of the denominator is not zero. Thus: \(\lim _ { x \rightarrow b } \frac { f ( x ) } { g ( x ) } = \frac{ \lim _ { x \rightarrow b } f ( x ) }{ \lim _ { x \rightarrow b } g ( x ) } = \frac{7}{-3} = -\frac{7}{3}\).