Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

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

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

Step 7.2

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

$\frac{2 \cdot 1 + y}{1 + 1} = 5$

Step 8

Simplify the answer.

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Step 8.1

Multiply $2$ by $1$ .

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

Step 8.2

Add $1$ and $1$ .

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