Solve for a limit as x approaches 1 of (2x+a)/(x+1)=5

$lim_{x \to 1} \frac{2 x + a}{x + 1} = 5$

Simplify $lim_{x \to 1} \frac{2 x + a}{x + 1}$ .

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 2 x + a}{lim_{x \to 1} x + 1} = 5$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 2 x + lim_{x \to 1} a}{lim_{x \to 1} x + 1} = 5$

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 1} x + lim_{x \to 1} a}{lim_{x \to 1} x + 1} = 5$

Evaluate the limit of $a$ which is constant as $x$ approaches $1$ .

$\frac{2 lim_{x \to 1} x + a}{lim_{x \to 1} x + 1} = 5$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{2 lim_{x \to 1} x + a}{lim_{x \to 1} x + lim_{x \to 1} 1} = 5$

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{2 lim_{x \to 1} x + a}{lim_{x \to 1} x + 1} = 5$

Simplify the numerator.

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{2 \cdot 1 + a}{lim_{x \to 1} x + 1} = 5$

Multiply $2$ by $1$ .

$\frac{2 + a}{lim_{x \to 1} x + 1} = 5$

Simplify the denominator.

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{2 + a}{1 + 1} = 5$

Add $1$ and $1$ .

$\frac{2 + a}{2} = 5$

Multiply both sides of the equation by $2$ .

$2 \frac{2 + a}{2} = 2 \cdot 5$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{2 + a}{2} = 2 \cdot 5$

Rewrite the expression.

$2 + a = 2 \cdot 5$

Simplify the right side.

Multiply $2$ by $5$ .

$2 + a = 10$

Move all terms not containing $a$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$a = 10 - 2$

Subtract $2$ from $10$ .

$a = 8$