Solving Limits with Functions: \lim_{t\rightarrow k}

Therefore, the limit of the quotient is \frac{7}{7}=1. In summary, when dealing with limits of functions, if both limits exist, the limit of the product is the product of the limits and the limit of the quotient is the quotient of the limits, as long as the limit on the bottom is nonzero. Additionally, the limit of a sum is just the sum of the limits, if both limits exist.
  • #1
tandoorichicken
245
0
I forgot how to do these kind of problem:

If [tex]\lim_{t\rightarrow k} F(t) = 7 [/tex] and [tex]\lim_{t\rightarrow k} G(t) = 0[/tex], then what is [tex]\lim_{t\rightarrow k} F(t)G(t)[/tex]?

Also:

What is [tex]\lim_{t\rightarrow k} \frac{F(t)}{G(t)+7}[/tex]?
 
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  • #2
Since both the limits exist, the limit of the product is just the product of the limits. Also, if you have a quotient and the limit on the top exists and the limit on the bottom exists and is nonzero, then the limit of the quotient is just the quotient of the limits.
If [tex]\lim_{t\rightarrow k} G(t)=0[/tex], then [tex]\lim_{t\rightarrow k} G(t)+7=7[/tex], since the limit of a sum is just the sum of the limits (if both limits exist).
 
  • #3



Don't worry, solving limits with functions can be a bit tricky at first but with practice, you'll get the hang of it again. To solve the first problem, we can use the limit laws which state that the limit of a product is equal to the product of the limits. So, in this case, we can say that \lim_{t\rightarrow k} F(t)G(t) = \lim_{t\rightarrow k} F(t) \cdot \lim_{t\rightarrow k} G(t) = 7 \cdot 0 = 0. Therefore, the limit of the product is 0.

For the second problem, we can use another limit law which states that the limit of a quotient is equal to the quotient of the limits, as long as the limit of the denominator is not 0. In this case, the limit of the denominator is 0, so we cannot use this law. Instead, we can try to factor out a common factor from the numerator and denominator to eliminate the 0 in the denominator.

\lim_{t\rightarrow k} \frac{F(t)}{G(t)+7} = \lim_{t\rightarrow k} \frac{F(t)}{G(t)} \cdot \frac{1}{1+\frac{7}{G(t)}}

Since we know that \lim_{t\rightarrow k} F(t) = 7 and \lim_{t\rightarrow k} G(t) = 0, we can rewrite this as:

= \frac{7}{0+7} = 1

Therefore, the limit is equal to 1. Keep practicing and don't be afraid to ask for help if you get stuck. Good luck!
 

FAQ: Solving Limits with Functions: \lim_{t\rightarrow k}

What is a limit in calculus?

A limit in calculus is a fundamental concept that describes the behavior of a function as the input approaches a certain value. It represents the value that the function approaches, but may not necessarily reach, as the input gets closer and closer to the given value.

How do you evaluate a limit using a function?

To evaluate a limit using a function, you can use direct substitution, algebraic manipulation, or graphical analysis. Direct substitution involves plugging in the given value for the input and solving for the resulting output. Algebraic manipulation involves simplifying the function to make it easier to evaluate. Graphical analysis involves looking at the graph of the function and determining the behavior as the input approaches the given value.

What does \lim_{t\rightarrow k} represent?

The notation \lim_{t\rightarrow k} represents the limit of a function as the input, t, approaches the value k. It can also be read as "the limit as t approaches k" or "the limit of f(t) as t approaches k".

Can a limit exist if the function is not defined at the given value?

Yes, a limit can exist even if the function is not defined at the given value. This is known as a one-sided limit, where the input approaches the given value from one side only. However, for the limit to exist, the function must approach the same value from both sides.

How do you prove the existence of a limit using the epsilon-delta definition?

The epsilon-delta definition is a rigorous method for proving the existence of a limit. It involves setting a range of values, or an epsilon, around the expected limit value and finding a corresponding range of inputs, or a delta, that will guarantee the function will output values within the epsilon range. If such a delta exists, the limit is proved to exist.

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