Question

1) lim (x→бесконечность) 100x-200x^2/400x^2+500x 2)lim(x→бесконечность) 3+x^3-x^5/x^2+20-x^4

Answer 1

Ответ:

В первом остаётся -0.5

Во втором бесконечность

Answer 2

To evaluate the given limits, let's simplify the expressions and apply the concept of limits as x approaches infinity.

1. lim (x→∞) (100x - 200x^2) / (400x^2 + 500x)

Step 1: Factor out x^2 from the numerator and x from the denominator to simplify the expression: lim (x→∞) x^2 * (100 - 200/x) / (x^2 * (400 + 500/x))

Step 2: As x approaches infinity, both (200/x) and (500/x) approach 0. Therefore, the expression becomes: lim (x→∞) x^2 * (100 - 0) / (x^2 * (400 + 0))

Step 3: Simplify further: lim (x→∞) (100x^2) / (400x^2) = 100/400 = 1/4

So, the first limit is 1/4.

2. lim (x→∞) (3 + x^3 - x^5) / (x^2 + 20 - x^4)

Step 1: As x approaches infinity, the highest powers of x in the numerator and denominator dominate the expression. The term x^5 in the numerator and the term x^4 in the denominator are the most significant.

Step 2: Factor out x^5 from the numerator and x^4 from the denominator to simplify the expression: lim (x→∞) x^5 * (3/x^5 + 1 - 1/x^5) / (x^4 * (1/x^2 + 20/x^4 - 1))

Step 3: As x approaches infinity, both (3/x^5) and (1/x^5) approach 0, and (1/x^2) and (20/x^4) also approach 0. The expression becomes: lim (x→∞) x^5 * (0 + 1 - 0) / (x^4 * (0 + 0 - 1))

Step 4: Simplify further: lim (x→∞) x^5 / (-x^4) = -x

So, the second limit is -∞.