answer:To solve the problem of finding the limit:[lim_{x rightarrow 0} frac{cos x + ln(1 + x) sqrt{2 + cos left(frac{1}{x}right)}}{2 + e^x}]we will follow these steps:1. Understand the Behavior of the Function as (x to 0): - Evaluate the numerator (cos x + ln(1 + x) sqrt{2 + cos left(frac{1}{x}right)}) as (x to 0). - Evaluate the denominator (2 + e^x) as (x to 0).2. Use Series Expansion or Simplification: - Use Taylor series expansion for the numerator and denominator around (x = 0), if necessary. - Simplify the expression to see if the limit can be evaluated directly. 3. Evaluate the Limit: - Calculate the limit using sympy.Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the functionnumerator = sp.cos(x) + sp.ln(1 + x) * sp.sqrt(2 + sp.cos(1/x))denominator = 2 + sp.exp(x)function = numerator / denominator# Calculate the limit as x approaches 0limit_value = sp.limit(function, x, 0)print(limit_value)``````output1/3```The limit of the function [lim_{x rightarrow 0} frac{cos x + ln(1 + x) sqrt{2 + cos left(frac{1}{x}right)}}{2 + e^x}]is (boxed{frac{1}{3}}).

answer:Let's break down each of the given expressions and calculate them step-by-step using Python and SymPy. 1. Expression: ((-xy^{2}z^{3}) cdot (-x^{2}y)^{3})First, simplify the expression using algebraic rules. 2. Expression: (-2a^{2}left(frac{1}{2}ab + b^{2}right))Distribute the (-2a^{2}) inside the parentheses. 3. Expression: (frac{(3a + b)^{2} - b^{2}}{a})Simplify the numerator and then divide by (a). 4. Expression: (frac{(x + y)^{2} - (x - y)^{2}}{2xy})Simplify the numerator and then divide by (2xy).Let's implement the simplification using SymPy in Python:```pythonimport sympy as sp# Define the symbolsx, y, z, a, b = sp.symbols('x y z a b')# Expression 1: (-xy^2z^3) * (-x^2y)^3expr1 = (-x*y2*z3) * (-x2*y)3simplified_expr1 = sp.simplify(expr1)# Expression 2: -2a^2(1/2ab + b^2)expr2 = -2*a2 * (1/2*a*b + b2)simplified_expr2 = sp.simplify(expr2)# Expression 3: [(3a + b)^2 - b^2] / aexpr3 = ((3*a + b)2 - b2) / asimplified_expr3 = sp.simplify(expr3)# Expression 4: [(x + y)^2 - (x - y)^2] / (2xy)expr4 = ((x + y)2 - (x - y)2) / (2*x*y)simplified_expr4 = sp.simplify(expr4)# Print the resultsprint(simplified_expr1)print(simplified_expr2)print(simplified_expr3)print(simplified_expr4)``````outputx7*y5*z3a2*b*(-1.0*a - 2*b)9*a + 6*b2```Let's go through each of the results step-by-step and ensure that they are correct:1. Expression: ((-xy^{2}z^{3}) cdot (-x^{2}y)^{3}) - Simplified Result: (x^7 y^5 z^3) - This is correct because: [ (-xy^2z^3) cdot (-x^2y)^3 = (-xy^2z^3) cdot (-x^6y^3) = x cdot y^2 cdot z^3 cdot x^6 cdot y^3 = x^7 y^5 z^3 ]2. Expression: (-2a^{2}left(frac{1}{2}ab + b^{2}right)) - Simplified Result: (a^2 b (-a - 2b)) - This is correct because: [ -2a^2left(frac{1}{2}ab + b^2right) = -2a^2left(frac{1}{2}abright) - 2a^2(b^2) = -a^3b - 2a^2b^2 = a^2b(-a - 2b) ]3. Expression: (frac{(3a + b)^{2} - b^{2}}{a}) - Simplified Result: (9a + 6b) - This is correct because: [ frac{(3a + b)^2 - b^2}{a} = frac{9a^2 + 6ab + b^2 - b^2}{a} = frac{9a^2 + 6ab}{a} = 9a + 6b ]4. Expression: (frac{(x + y)^{2} - (x - y)^{2}}{2xy}) - Simplified Result: (2) - This is correct because: [ frac{(x + y)^2 - (x - y)^2}{2xy} = frac{(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)}{2xy} = frac{4xy}{2xy} = 2 ]Thus, the results are:1. (x^7 y^5 z^3)2. (a^2 b (-a - 2b))3. (9a + 6b)4. (2)The final answers are:[boxed{x^7 y^5 z^3}, quad boxed{a^2 b (-a - 2b)}, quad boxed{9a + 6b}, quad boxed{2}]

answer:To solve for ( x ) given the dot product condition (boldsymbol{a} cdot boldsymbol{b} = 1), we can break down the problem into the following steps:1. Understand the Dot Product: The dot product of two vectors (boldsymbol{a} = (a_1, a_2)) and (boldsymbol{b} = (b_1, b_2)) is given by: [ boldsymbol{a} cdot boldsymbol{b} = a_1 b_1 + a_2 b_2 ]2. Apply Given Values: Given (boldsymbol{a} = (1, -1)) and (boldsymbol{b} = (2, x)), the dot product is: [ boldsymbol{a} cdot boldsymbol{b} = 1 cdot 2 + (-1) cdot x = 2 - x ] We are given that this dot product equals 1: [ 2 - x = 1 ]3. Solve for ( x ): Solve the equation (2 - x = 1) for ( x ).Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the equation from the dot product conditionequation = 2 - x - 1# Solve the equationsolution = sp.solve(equation, x)# Output the solutionprint(solution[0])``````output1```The solution to the equation (2 - x = 1) is (x = 1).Therefore, ( boxed{1} ) is the value of ( x ) that satisfies the given condition.

answer:To solve the problem, we need to determine the sum of coefficients ( a_2 + a_3 + cdots + a_9 + a_{10} ) in the expansion of ((2x - 1)^{10}).We can use the binomial theorem to expand ((2x - 1)^{10}), which states:[(2x - 1)^{10} = sum_{k=0}^{10} binom{10}{k} (2x)^k (-1)^{10-k}]The general term in the expansion is:[a_k x^k = binom{10}{k} (2x)^k (-1)^{10-k}]To find the sum ( a_2 + a_3 + cdots + a_9 + a_{10} ), we will sum the coefficients of the respective terms. Note that the coefficient of ( x^k ) is given by:[a_k = binom{10}{k} 2^k (-1)^{10-k}]Let's compute the sum using Python and SymPy:```pythonimport sympy as sp# Define the binomial coefficient functiondef binomial_coeff(n, k): return sp.binomial(n, k)# Define the sum of coefficients a_2 + a_3 + ... + a_9 + a_10n = 10coeff_sum = 0for k in range(2, 11): coeff = binomial_coeff(n, k) * 2k * (-1)(n-k) coeff_sum += coeffprint(coeff_sum)``````output20```The sum of the coefficients (a_2 + a_3 + cdots + a_9 + a_{10}) in the expansion of ((2x - 1)^{10}) is (boxed{20}).The sum of the coefficients (a_2 + a_3 + cdots + a_9 + a_{10}) in the expansion of ((2x - 1)^{10}) is (boxed{20}).

answer:To solve this problem, we need to find prime numbers ( p ) such that ( p ) divides ( 2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1} ). We'll iterate over prime numbers and check the condition for each. Steps:1. Generate a list of prime numbers.2. For each prime number ( p ), compute ( 2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1} ).3. Check whether ( p ) divides this sum.4. Keep track of the largest prime ( p ) that satisfies the condition.Let's start by writing Python code to perform these steps.```pythonimport sympy # Function to find the largest prime that divides the given expression def find_largest_prime(divisor_limit): largest_prime = 0 prime_list = list(sympy.primerange(1, divisor_limit)) # Generate primes up to divisor_limit for p in prime_list: expression_value = 2(p+1) + 3(p+1) + 5(p+1) + 7(p+1) if expression_value % p == 0: largest_prime = p return largest_prime # Set a reasonable limit for prime checking (it can be increased) divisor_limit = 1000 largest_prime = find_largest_prime(divisor_limit) print(largest_prime)``````output29```The largest prime ( p ) that divides ( 2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1} ) within the checked range ( {1, ldots, 1000} ) is:(boxed{29})

answer:To identify the smallest abundant number, we need to:1. Define a function to calculate the sum of the proper divisors of a number.2. Check numbers sequentially to find the smallest number where the sum of its proper divisors is greater than the number itself.First, let's define the function to calculate the sum of a number's proper divisors. Proper divisors of a number ( n ) are the divisors of ( n ) excluding ( n ) itself. Then we can iterate through the natural numbers to find the smallest abundant number.Let's write the code to accomplish this.```pythondef sum_of_proper_divisors(n): divisors_sum = 1 # 1 is a proper divisor of all natural numbers sqrt_n = int(n 0.5) for i in range(2, sqrt_n + 1): if n % i == 0: divisors_sum += i if i != n // i: # add the quotient only if it's different from i divisors_sum += n // i return divisors_sumdef find_smallest_abundant_number(): n = 1 while True: if sum_of_proper_divisors(n) > n: return n n += 1smallest_abundant_number = find_smallest_abundant_number()print(smallest_abundant_number)``````output12```The smallest abundant number is ( boxed{12} ).