Example Question - limit calculation
Here are examples of questions we've helped users solve.
Calculating Limit of a Trigonometric Expression
Para encontrar el límite de la expresión \(\lim_{{x \to 0}} \frac{{\sin^3 x}}{{x^2}}\), podemos utilizar la conocida regla del límite: \(\lim_{{x \to 0}} \frac{{\sin x}}{{x}} = 1\). Sin embargo, la expresión que tenemos es un poco diferente y tenemos que ajustarla para poder aplicar directamente esta regla. Primero, podemos reescribir el seno al cubo como \((\sin x)^3\). Entonces, la expresión queda como \((\sin x)^3/x^2\), lo cual puede ser escrita como \((\sin x/x)^2 \cdot \sin x\). Ahora aplicamos la regla mencionada: \[\lim_{{x \to 0}} \frac{{\sin x}}{{x}} = 1\] Utilizamos esto para calcular el valor del límite: \[\lim_{{x \to 0}} (\frac{{\sin x}}{{x}})^2 \cdot \sin x = (\lim_{{x \to 0}} \frac{{\sin x}}{{x}})^2 \cdot (\lim_{{x \to 0}} \sin x)\] Sustituimos los límites conocidos: \[= 1^2 \cdot 0\] Así que el resultado del límite es: \[= 0\] Por lo tanto, el valor del límite es 0.
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.
Understanding Limits of a Function from a Graph
The image shows a graph of a function y = f(x) and asks for the values of various limits as x approaches different numbers: a) \( \lim_{x \to -2^-} f(x) \) b) \( \lim_{x \to -2^+} f(x) \) c) \( \lim_{x \to -2} f(x) \) d) \( \lim_{x \to 3} f(x) \) e) \( \lim_{x \to +\infty} f(x) \) Based on the graph depicted in the image: a) \( \lim_{x \to -2^-} f(x) \) is the limit of the function as x approaches -2 from the left. The graph shows that as x approaches -2 from the left, y approaches -1. So the limit is -1. b) \( \lim_{x \to -2^+} f(x) \) is the limit of the function as x approaches -2 from the right. The graph shows that as x approaches -2 from the right, y approaches +1. So the limit is +1. c) \( \lim_{x \to -2} f(x) \) is the limit of the function as x approaches -2 from both the left and right. Since the limits from the left and right are different, the overall limit does not exist. So the limit is undefined or does not exist. d) \( \lim_{x \to 3} f(x) \) is the limit of the function as x approaches 3 from either direction. The graph shows that the y-values are approaching a value around +2. So the limit is +2. e) \( \lim_{x \to +\infty} f(x) \) is the limit of the function as x approaches infinity. The graph shows that as x goes to infinity (to the right), y is approaching a value of -3. So the limit is -3.
Limit Calculation of Rational Function
The limit given in the image is \[ \lim_{{x \to 5}} \frac{x - 5}{x^2 - 25} \] We can start by factoring the denominator: \[ x^2 - 25 = (x - 5)(x + 5) \] The expression for the limit then becomes: \[ \lim_{{x \to 5}} \frac{x - 5}{(x - 5)(x + 5)} \] We can simplify the expression by canceling out the common factor of \(x - 5\) in the numerator and the denominator: \[ \lim_{{x \to 5}} \frac{1}{x + 5} \] Now we can directly substitute \(x = 5\) into the simplified expression, as there are no more discontinuities: \[ \frac{1}{5 + 5} = \frac{1}{10} \] So, the value of the limit is \(\frac{1}{10}\).
Piecewise Graph Limit Calculation
The image presents a piecewise graph of the function y = f(x) and asks for the value of the expression: \[ \lim_{x \to -4^+} f(x) - \lim_{x \to 0^-} f(x) + \lim_{x \to 2^+} f(x) \] From the graph, we can derive the following limits: 1. \( \lim_{x \to -4^+} f(x) \) is the value that f(x) approaches as x approaches -4 from the right. By looking at the graph, we can see this value is n. 2. \( \lim_{x \to 0^-} f(x) \) is the value that f(x) approaches as x approaches 0 from the left. On the graph, this value is m. 3. \( \lim_{x \to 2^+} f(x) \) is the value that f(x) approaches as x approaches 2 from the right. The graph shows this value as k. Now, we substitute these values into the given expression: n (the limit as x approaches -4 from the right) - m (the limit as x approaches 0 from the left) + k (the limit as x approaches 2 from the right). From the graph, we can infer the following based on the labelled points: - n is at y = -1 - m is at y = 1 - k is at y = 0 Using these values: \( \lim_{x \to -4^+} f(x) = -1 \) \( \lim_{x \to 0^-} f(x) = 1 \) \( \lim_{x \to 2^+} f(x) = 0 \) Substitute these values into the expression: \[ (-1) - (1) + (0) = -2 \] Therefore, the value of the given expression is -2.
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