[ASK] Seemingly Simple Limit Question but I have no Idea

In summary, the value of the limit is 5. L'Hopital's rule is used to find the derivative of the numerator and denominator, and the given values of f(a), f'(a), g(a), and g'(a) are used to calculate the limit.
  • #1
Monoxdifly
MHB
284
0
If f(a) = 2, f'(a) = 1, g(a) = –1, and g'(a) = 2, the value of \(\displaystyle \lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a}\) is ...
A. 1
B. 3
C. 5
D. 7
E. 9

\(\displaystyle \lim_{x\to a}\frac{g(x)\cdot f(a)-g(a)\cdot f(x)}{x-a}=\lim_{x\to a}\frac{2g(x)+f(x)}{x-a}\). How to determine the f(x) and g(x)? And when to use the info that f'(a) = 1 and g'(a) = 2?
 
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  • #2
You CAN'T "determine the f(x) and g(x)" and you don't need to do. That is not asked.

Taking x= a gives 0/0 so we can use "L'Hopitals rule". The derivative of the numerator is g'(x)f(a)- g(a)f'(x) and the derivative of the denominator is 1 so the limit is $\lim_{x\to a} g'(x)f(a)- g(a)f'(x)= g'(a)f(a)-g(a)f'(a)= (2)(2)- (1)(-1)= 4+ 1= 5$.
 
  • #3
Ah, so it is indeed a simple question. My bad, Thanks for your help. :)
 

FAQ: [ASK] Seemingly Simple Limit Question but I have no Idea

What is a limit in mathematics?

A limit is a fundamental concept in mathematics that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value.

What is the purpose of finding limits?

The purpose of finding limits is to understand the behavior of a function and to determine the value that a function approaches as its input gets closer and closer to a specific value. It is also used to solve various problems in calculus and other branches of mathematics.

How do you solve seemingly simple limit questions?

To solve seemingly simple limit questions, you need to first understand the basic principles of limits and the various techniques for evaluating them. You also need to have a strong understanding of algebra and calculus to apply these techniques effectively. Practice and familiarity with different types of limit problems is also important.

What are some common mistakes when solving limit questions?

Some common mistakes when solving limit questions include not applying the correct limit rules, not simplifying the expression before evaluating the limit, and not considering the behavior of the function at the specific value. It is also important to check for any discontinuities or points of discontinuity in the function.

How can I improve my skills in solving limit questions?

To improve your skills in solving limit questions, it is important to practice regularly and familiarize yourself with different types of limit problems. You can also seek help from a tutor or attend a study group to discuss and solve limit problems together. Additionally, reviewing and understanding the basic principles of limits and the various techniques for evaluating them can also help improve your skills.

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