Final Answer:
The limit of (x²+4)/(5-3x) as x approaches -2 is undefined.
Step-by-step explanation:
To find the limit, substitute -2 for x in the given expression:
(−2)^2+4/5−3(−2). This simplifies to 5+6/4+4, which is 8/11. However, since the denominator approaches zero (5 + 6 = 11) as x approaches -2, the limit is undefined. In such cases, the expression exhibits a vertical asymptote at x = -2, indicating that the function approaches infinity or negative infinity as x approaches -2 from the left or right.
Understanding limits is essential in analyzing the behavior of a function as it approaches a particular point. In this case, the limit is undefined due to the denominator approaching zero, leading to an indeterminate form (0/0). The presence of a vertical asymptote at x = -2 indicates a discontinuity in the function at this point, and further analysis, such as factoring or simplification, may be necessary to determine the limit accurately.
In conclusion, the limit of (x²+4)/(5-3x) as x approaches -2 is undefined, signifying a vertical asymptote at x = -2. This mathematical concept is crucial in understanding the behavior of functions near specific points and aids in identifying points of discontinuity or divergence.