Hi everyone, today we're diving into the topic of limits in calculus. It can be a bit tricky, but with practice and dedication, anyone can master it. Let's take a look at some practice problems that will help sharpen your skills.
Practice Problem #1
In this problem, we're asked to evaluate the limit of the function (3x+2)/(2x-1) as x approaches 2. To solve this problem, we need to substitute 2 for x in the function and simplify:
(3(2)+2)/(2(2)-1) = 8/3
Therefore, the limit of the function as x approaches 2 is 8/3.
Practice Problem #2
This problem asks us to evaluate the limit of the function (x^2-4)/(x-2) as x approaches 2. To solve this problem, we need to factor the polynomial in the numerator:
(x-2)(x+2)/(x-2) = x+2
Therefore, the limit of the function as x approaches 2 is 4.
Practice Problem #3
In this problem, we're asked to evaluate the limit of the function (x^2-4)/(x-2) as x approaches -2. To solve this problem, we again need to factor the polynomial in the numerator:
(x-2)(x+2)/(x-2) = x+2
However, we can't simply substitute -2 for x, because then we'd be dividing by zero (since x-2 would equal -4). Instead, we need to approach -2 from both sides of the limit:
As x approaches -2 from the left (i.e. values of x that are slightly greater than -2), the value of x+2 approaches 0:
(-2+0)/(1) = -2
As x approaches -2 from the right (i.e. values of x that are slightly less than -2), the value of x+2 also approaches 0:
(-2+0)/(1) = -2
Therefore, the limit of the function as x approaches -2 is -2.
Keep practicing these types of problems and you'll soon become a limit-solving master!
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