Find the limit
lim
x->2
(2x^2 +1)/(x^2 + 6x -4)
step 1 in any limit question is to actually substitute the approach value into your expression.
If you get a real number, that is your answer, you are done!
lim (2x^2 + 1)/(x^2 + 6x - 4) as x --> 2
= 9/12
= 3/4
Voila!!! all done.
To find the limit as x approaches 2 for the given function:
lim(x->2) (2x^2 + 1)/(x^2 + 6x - 4)
We can directly substitute the value of 2 into the function:
(2(2^2) + 1)/(2^2 + 6(2) - 4)
Simplifying this expression further:
(2(4) + 1)/(4 + 12 - 4)
(8 + 1)/(20)
9/20
Therefore, the limit as x approaches 2 for the given function is 9/20.
To find the limit of the given function as x approaches 2, we need to evaluate the function at x = 2. However, if we directly substitute 2 into the function, we would get an undefined form (0/0), which is not helpful for finding the limit.
To overcome this, we can use algebraic manipulation to simplify the given function and then substitute the desired value. Here's how you can do it:
Step 1: Simplify the function by factoring the denominator.
The denominator is a quadratic expression and can be factored as (x + 4)(x - 2).
(2x^2 + 1)/((x + 4)(x - 2))
Step 2: Cancel out common factors.
We can cancel out the common factor (x - 2) in the numerator and denominator.
= (2x^2 + 1)/(x + 4)
Step 3: Substitute the value of x.
Now that we have a simplified expression, we can substitute the value of x, which is 2.
= (2(2)^2 + 1)/(2 + 4)
= (2(4) + 1)/(6)
= (8 + 1)/(6)
= 9/6
Step 4: Simplify the result.
To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 3.
= (9 ÷ 3)/(6 ÷ 3)
= 3/2
Therefore, the limit as x approaches 2 of the given function is 3/2.