Example Question - horizontal asymptote
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Understanding a Function's Graph
La imagen muestra la gráfica de una función f(x) que está definida como: \[ f(x) = \frac{x}{x^2 - 8x + 16} \] Para comprender la gráfica, primero debemos simplificar y entender la expresión algebraica dada. Observa que el denominador \( x^2 - 8x + 16 \) es un trinomio cuadrado perfecto que se puede factorizar como \((x - 4)^2\). Así que la función se puede reescribir como: \[ f(x) = \frac{x}{(x - 4)^2} \] Dado que la gráfica de la función ya está dada en la imagen, describiremos las características más importantes: 1. La función tiene una asíntota vertical en x = 4, que es el valor para el cual el denominador se hace cero y la función se indefine. En el gráfico se observa que la función crece indefinidamente a medida que x se aproxima a 4 desde la izquierda y decrece indefinidamente a medida que x se aproxima a 4 desde la derecha. 2. La función tiene una asíntota horizontal en y = 0 cuando x tiende a infinito o a menos infinito, ya que el grado del polinomio en el denominador es mayor que el grado del numerador. La gráfica se aproxima a la línea y = 0 pero nunca la toca. 3. La función cruza el eje de las ordenadas (eje y) en el punto (0, 0), ya que f(0) = 0. 4. La función es positiva para valores de x menores que 4 y mayores que 4, esto es reflejado por el hecho de que la gráfica está por encima del eje x en esos intervalos. 5. La función es simétrica respecto al eje y porque el numerador es x (una función impar) y el denominador es una función par ((x - 4)^2). Esto resulta en una función impar, lo que explica por qué la gráfica es simétrica respecto al origen. Si tienes alguna otra pregunta específica sobre la gráfica o la función, no dudes en preguntar.
Limit Calculation on Function Graph
The image shows a function graphed on a coordinate plane, and there are several limits that need to be calculated. While I cannot interact with the image to enlarge it or click on it, I can analyze it based on the image provided. 1. The first limit is \(\lim_{x\to c^-} f(x)\), where `c` is the x-value at the vertical asymptote where the graph tends towards infinity as x approaches from the left. From the image, the graph shoots upwards towards positive infinity as x approaches this value from the left side. 2. The second limit is \(\lim_{x\to c^+} f(x)\), for the same value `c` (the vertical asymptote). From the graph, when approaching from the right side of the vertical asymptote, the function appears to head downwards towards negative infinity. 3. The third limit is \(\lim_{x\to d^-} f(x)\), where `d` represents the x-value where the function is heading toward as x approaches from the left. In this case, the graph approaches a horizontal asymptote from the left. However, without precise values or markings on the graph, I cannot indicate what the value of the limit is, so I can generally say that the limit has a certain finite value, which we can call 'L', where \(L\) is the y-value of the horizontal asymptote. 4. The fourth limit is \(\lim_{x\to d^+} f(x)\), which is as x approaches `d` from the right. From the graph, it seems the function approaches the same horizontal asymptote as it does from the left, thus also appearing to approach the same value 'L'. To summarize, the answers based on the graph (using general terms since specific values are not given): 1. \(\lim_{x\to c^-} f(x) = \infty\) 2. \(\lim_{x\to c^+} f(x) = -\infty\) 3. \(\lim_{x\to d^-} f(x) = L\) 4. \(\lim_{x\to d^+} f(x) = L\) Please bear in mind that 'L' stands for the finite limit value at the horizontal asymptote. In a precise solution, you would need to replace this with the actual y-coordinate value of the horizontal asymptote.
Limits Analysis of a Function Graph
The image shows a graph of a function \( y = f(x) \) with various labeled points and behaviors around certain x-values, which we are asked to analyze to determine the limits. a) \( \lim_{x \to -3^-} f(x) = \) The graph approaches the y-value as x approaches -3 from the left side (negative side), which looks to be \( -\infty \). b) \( \lim_{x \to -3^+} f(x) = \) When x approaches -3 from the right side (positive side), the graph approaches a finite y-value, which appears to be 2. c) \( \lim_{x \to 2^-} f(x) = \) As x approaches 2 from the left side, the graph approaches the y-value which appears to be 3. d) \( \lim_{x \to 2^+} f(x) = \) When x approaches 2 from the right side, the graph is again approaching the y-value of 3. e) \( \lim_{x \to \infty} f(x) = \) As x goes to infinity, the y-value that the graph tends toward seems to be 0, which represents a horizontal asymptote at y = 0. So the answers are: a) \( -\infty \) b) 2 c) 3 d) 3 e) 0
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